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Alain 8adiou + Number and Numbers
Number and Numbers
Alain Badiou
Translated by Robin Mackay
First published in French as Le Nombre et les lIombres <0 Editions du Seuil,
This English edition © Polity Press, 2008
Reprinted in 2008
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Translator's Preface vii
0 Number Must Be Thought
Genealogies: Frege, Dedekind, Peano,
Cantor 5
I Greek Number and Modern Number 7
2 Frege 16
3 Additional Note on a Contemporary
Usage of Frege 24
4 Dedekind 31
5 Peano 46
6 Cantor: 'Well-Ordered ness' and the
Ordinals 52
2 Concepts: Natural Multiplicities 59
7 Transitive Multiplicities 61
8 Von Neumann Ordinals 67
9 Succession and Limit. The Infinite 73
10 Recurrence, or Induction 83
II Natural Whole Numbers 93
vi CONTENTS
3 Ontology of Number: Definition, Order,
Cuts, Types 99
12 The Concept of Number: An Evental
Nomination 101
13 Difference and Order of Numbers 115
14 The Concept of Sub-Number 131
15 Cuts: The Fundamental Theorem 139
16 The Numberless Enchantment of the
Place of Number 156
4 Operational Dimensions 181
17 Natural Interlude 183
18 Algebra of Numbers 198
Conclusion 209
19 In Conclusion: From Number to Trans-Being 211
Notes 215
Index 231
Translator's Preface
Alain Badiou's Number and Numbers, first published two years after
his Being and Event, is far from being the specialist work its title
might suggest. In fact, it recapitulates and deepens Being and Event's
explosion of the pretexts upon which the 'philosophy of mathematics'
is reduced to a theoretical ghetto; and their kinship to those reaction-
ary modes of thought that systematically obscure the most pressing
questions for contemporary philosophy. Neither does Number and
Numbers balk at suggesting that even the greatest efforts on the part
of number-theorists themselves have fallen short of the properly
radical import of the question of number. Badiou's astonishing analy-
ses in the historical section of the book uncover the inextricable bond
between philosophical assumptions and mathematical approaches to
the problem in these supposedly 'merely technical' works. The aim
of Number and Numbers, then, is certainly not to mould the unwill-
ing reader into a calculating machine, or a 'philosopher of mathemat-
ics': its exhortation is that we (mathematicians, philosophers, subjects
under Capital) systematically think number out of the technical,
procedural containment of which its quotidian tyranny, and the
abysmal fear it strikes into the heart of the non-mathematician, are
but symptoms. Symptoms, needless to say, whose expression within
the situation of philosophy is a pronounced distaste for number-as-
philosopheme - whence its recognisable absence in much 'continental
philosophy', except where it is pilloried as the very nemesis of the
ontological vocation. So if the 'return of the numerical repressed'
proposed here will, by definition, excite a symptomatic resistance, for
viii TRANSLATOR'S PREFACE
Badiou it alone can clear the way for the proper task of philosophy;
as a working-through of the mathematical ontology presented in
Being and Eve1Zt, Number and Numbers is a thorough conceptual
apprenticeship preparatory to the thinking of the event.
For the great thinkers of number-theory at the end of the nine-
teenth century, the way to an ontological understanding of number
was obscured by calculatory and operational aspects. Today, accord-
ing to Badiou, the political domination of number under capitalism
demands that the project be taken up anew: only if contemporary
philosophy rigorously thinks through number can it hope to cut
through the apparently dense and impenetrable capitalist fabric of
numerical relations, to think the event that can 'subtract' the subject
from that 'ontic' skein without recourse to an anti-mathematical
Whilst this doubtless demands 'one more effort' on the part of the
non-mathematician, it would be a peevish student of philosophy who,
understanding the stakes and contemplating the conceptual vista
opened up, saw this as an unreasonable demand - especially when
Badiou offers to those lacking in mathematical knowledge the rare
privilege of taking a meticulously navigated conceptual shortcut to
the heart of the matter.
Badiou's remarkable book comprises a number of different works
- a radical philosophical treatise, a contribution to number-theory, a
document in the history of mathematics, a congenial textbook and a
subtle and subversive exercise in political theory - whose intricate
interdependencies defy any order of priority. The translator's task is
to reproduce, with a foreign tongue, that unique voice that can
compel us to 'count as one' these disparate figures. In negotiating this
challenge, I have sought to prioritise clarity over adherence to any
rigid scheme of translation, except where mathematical terminology
demands consistent usage, or where an orthodoxy is clearly already
in force within extant translations of Badiou's work. In the latter case,
my references have been Oliver Feltham's landmark translation of
Being and Event, I with which I have sought to harmonise key terms,
Peter Hallward's invaluable A Subject to Truth/ and Ray Brassier
and Alberto Toscano's collection of Badiou's Theoretical Writings.
Apart from these, in translating chapters 2 and 3 I referred closely
to Sam Gillespie and Justin Clemens' translation in UMBR(a), Science
and Truth (2000). Finally, whilst seeking also to maintain continuity
with long-standing English translations of number-theoretical works,
some classics in their own right, occasionally the rigour of Badiou's
thinking has demanded a re-evaluation of their chosen translations
for key terms.
Translators also find themselves obliged to arbitrate
TRANSLATOR'S PREFACE ix
between a fidelity to Badiou's in many ways admirable indifference
to the pedantic apparatus of scholarly citation, and the temptation
to pin down the allusions and quotations distributed throughout his
work. Badiou's selection of texts is so discerning, however, that it is
hardly a chore to return to them. Having thus had frequent recourse
to the texts touched on in Number and Numbers (particularly in the
first, historical part), I have seen no reason not to add citations where
One presumes that those self-conscious styles of philosophical
writing that necessitate laboured circumlocutions or terminological
preciosity on the part of a translator would for Badiou fall under the
sign of 'modern sophistry', taken to task herein, as elsewhere in his
work. Nevertheless, the aspiration to universal conceptual transpar-
ency does not preclude consideration of Badiou as stylist: firstly, as
Oliver Feltham has remarked, Badiou's sentences utilise subject/verb
order in a characteristic way, and I have retained his tensile syntax
whenever doing so does not jeopardise comprehension in translation.
Perhaps just as importantly, Badiou does not achieve the deft and
good-humoured development of such extremely rich and complex
conceptual structures as are found in Number and Numbers without
a generous and searching labour on behalf of the reader, not to
mention a talent for suspense. Although the later sections of Number
and Numbers may seem daunting, I hope to have reproduced Badiou's
confident, meticulous, but never stuffy mode of exposition so as to
ease the way as much as possible. In fact, in contrast to his own
occasionally chilly edicts, I would venture to suggest that here, 'in his
element', Badiou allows himself a certain enthusiasm. One certainly
does not accompany him on this odyssey without also developing a
taste for the 'bitter joy' of Number.
This translation slowly came to fruition on the basis of a somewhat
impulsive decision; it may not have survived to completion without
the enthusiasm and aid of an internationally dispersed group of
friends and acquaintances, actual and virtual, with whom I shared
the work in progress. I would like to extend my thanks to those who
helped by pointing out errors and offering advice on the evolving
manuscript: Anindya Bhattacharyya, Ray Brassier, Michael Carr,
Howard Caygill, Thomas Duzer, Zachary L. Fraser, Peter Hallward,
Armelle Menard Seymour, Reza Negarestani, Robin Newton, Nina
Power, Manuela Tecusan, Alberto Toscano, Keith Tilford, David
Sneek, and Damian Veal. My thanks also to Alain Badiou for his
generous help and encouragement, and to the Institution and Staff of
the Bodleian, Taylor Institution, and Radcliffe Science Libraries in
Oxford. Part of my work on the translation was undertaken whilst
in receipt of a studentship from the Centre for Research in Modern
European Philosophy at Middlesex University, London.
My greatest debt of gratitude is to Ruth, without whose love and
understanding my battles with incomprehension could not even be
staged; and to Donald, a great inspiration, for whom the infinite joys
of number still lie ahead.
Robin Mackay
Number Must Be Thought
0.1. A paradox: we live in the era of number's despotism; thought
yields to the law of denumerable multiplicities; and yet (unless perhaps
this very default, this failing, is only the obscure obverse of a concept-
less submission) we have at our disposal no recent, active idea of
what number is. An immense effort has been made on this point, but
its labours were essentially over by the beginning of the twentieth
century: they are those of Dedekind, Frege, Cantor, and Peano. The
factual impact of number only escorts a silence of the concept. How
can we grasp today the question posed by Dedekind in his 1888
treatise, Was sind und was sollen die Zahlen?l We know very well
what numbers are for: they serve, strictly speaking, for everything,
they provide a norm for All. But we still don't know what they are,
or else we repeat what the great thinkers of the late nineteenth
century - anticipating, no doubt, the extent of their future jurisdiction
- said they were.
0.2. That number must rule, that the imperative must be: 'count!' -
who doubts this today? And not in the sense of that maxim which,
as Dedekind knew, demands the use of the original Greek when
retraced: &d 0 av8Qu)3to; - because it prescribes, for
thought, its singular condition in the matheme. For, under the current
empire of number, it is not a question of thought, but of realities.
0.3. Firstly, number governs our conception of the political, with
the currency - consensual, though it enfeebles every politics of the
thinkable - of suffrage, of opinion polls, of the majority. Every 'politi-
cal' convocation, whether general or local, in polling-booth or parlia-
ment, municipal or international, is settled with a count. And every
opinion is gauged by the incessant enumeration of the faithful (even
if such an enumeration makes of every fidelity an infidelity). What
counts - in the sense of what is valued - is that which is counted.
Conversely, everything that can be numbered must be valued. 'Politi-
cal Science' refines numbers into sub-numbers, compares sequences
of numbers, its only object being shifts in voting patterns - that is,
changes, usually minute, in the tabulation of numbers. Political
'thought' is a numerical exegesis.
0.4. Number governs the quasi-totality of the 'human sciences'
(although this euphemism can barely disguise the fact that what is
called 'science' here is a technical apparatus whose pragmatic basis
is governmental). Statistics invades the entire domain of these disci-
plines. The bureaucratisation of knowledges is above all an infinite
excrescence of numbering.
At the beginning of the twentieth century, sociology unveiled its
proper dignity - its audacity, even - in the will to submit the figure
of communitarian bonds to number. It sought to extend to the social
body and to representation the Galilean processes of literalisation
and mathematisation. But ultimately it succumbed to an anarchic
development of this enterprise. It is now replete with pitiful enumera-
tions that serve only to validate the obvious or to establish parlia-
mentary opportunities.
History has drawn massively upon statistical technique and is -
even, in fact above all, under the auspices of academic Marxism -
becoming a diachronic sociology. It has lost that which alone had
characterised it, since the Greek and Latin historians, as a discipline
of thought: its conscious subordination to the real of politics. Even
when it passes through the different phases of reaction to number -
economism, sociologism - it does so only to fall into their simple
inverse: biography, historicising psychologism.
And medicine itself, apart from its pure and simple reduction to its
scientific Other (molecular biology), is a disorderly accumulation of
empirical facts, a huge web of blindly tested numerical correlations.
These are 'sciences' of men made into numbers, to the saturation
point of all possible correspondences between these numbers and
other numbers, whatever they might be.
0.5. Number governs cultural representations. Of course, there is
television, viewing figures, advertising. But that's not the most
important thing. It is in its very essence that the cultural fabric is
woven by number alone. A 'cultural fact' is a numerical fact. And,
conversely, whatever produces number can be culturally located; that
which has no number will have no name either. Art, which deals with
number only in so far as there is a thinking of number, is a culturally
unpronounceable word.
0.6. Obviously, number governs the economy; and there, without a
doubt, we find what Louis Althusser would have called the 'determi-
nation in the last instance' of its supremacy. The ideology of modern
parliamentary societies, if they have one, is not humanism, law, or
the subject. It is number, the countable, countability. Every citizen is
expected to be cognisant of foreign trade figures, of the flexibility of
the exchange rate, of fluctuations in stock prices. These figures are
presented as the real to which other figures refer: governmental
figures, votes and opinion polls. Our so-called 'situation' is the inter-
section of economic numericality and the numericality of opinion.
France (or any other nation) can only be represented on the balance-
sheet of an import-export business. The only image of a country is
this inextricable heap of numbers in which, we are told, its power is
vested, and which, we hope, is deemed worthy by those who record
its mood.
0.7. Number informs our souls. What is it to exist, if not to give a
favourable account of oneself? In America, one starts by saying how
much one earns, an identification that is at least honest. Our old
country is more cunning. But still, you don't have to look far to dis-
cover numerical topics that everyone can identify with. No one can
present themselves as an individual without stating in what way they
count, for whom or for what they are really counted. Our soul has
the cold transparency of the figures in which it is resolved.
0.8. Marx: 'the icy water of egotistical calculation'.3 And how! To
the point where the Ego of egoism is but a numerical web, so that
the 'egotistical calculation' becomes the cipher of a cipher.
0.9. But we don't know what a number is, so we don't know what
we are.
0.10. Must we stop with Frege, Dedekind, Cantor or Peano? Hasn't
anything happened in the thinking of number? Is there only the
exorbitant extent of its social and subjective reign? And what sort
of innocent culpability can be attributed to these thinkers? To what
extent does their idea of number prefigure this anarchic reign? Did
they think number, or the future of generalised numericality? Isn't
another idea of number necessary, in order for us to turn thought
back against the despotism of number, in order that the Subject might
be subtracted from it? And has mathematics simply stood by silently
during the comprehensive social integration of number, over which
it formerly had monopoly? This is what I wish to examine.
Genealogies: Frege, Dedekind,
Peano, Cantor
Greek Number and
Modern Number
1.1. The Greek thinkers of number related it back to the One, which,
as we can still see in Euclid's Elements,
was considered not to be a
number. From the supra-numeric being of the One, unity is derived.
And a number is a collection of units, an addition. Underlying this
conception is a problematic that stretches from the Eleatics through
to the Neoplatonists: that of the procession of the Multiple from the
One. Number is the schema of this procession.
1.2. The modern collapse of the Greek thinking of number proceeds
from three fundamental causes.
The first is the irruption of the problem of the infinite - ineluctable
from the moment when, with differential calculus, we deal with
the reality of series of numbers which, although we may consider
their limit, cannot be assigned any terminus. How can the limit of
such a series be thought as number through the sole concept of
a collection of units? A series tends towards a limit: it is not the
addition of its terms or its units. It cannot be thought as a procession
of the One.
The second cause is that, if the entire edifice of number is sup-
ported by the being of the One, which is itself beyond being, it is
impossible to introduce, without some radical subversion, that other
principle - that ontological stopping point of number - which is zero,
or the void. It could be, certainly - and Neoplatonist speculation
appeals to such a thesis - that the ineffable and archi-transcendent
character of the One can be marked by zero. But then the problem
comes back to numerical one: how to number unity, if the One that
supports it is void? This problem is so complex that, as we shall see,
it remains today the key to a modern thinking of number.
The third reason, and the most contemporary one, is the pure and
simple dislocation of the idea of a being of the One. We find ourselves
under the jurisdiction of an epoch that obliges us to hold that being
is essentially multiple. Consequently, number cannot proceed from
the supposition of a transcendent being of the One.
1.3. The modern thinking of number thus found itself compelled to
forge a mathematics subtracted from this supposition. In so doing, it
took three different paths:
Frege's approach, and that of Russell (which we will call, for
brevity, the logicist approach), seeks to 'extract' number from a pure
consideration of the laws of thought itself. Number, according to this
point of view, is a universal trait
of the concept, deducible from
absolutely original principles (principles without which thought in
general would be impossible).
Peano's and Hilbert's approach (let's call this the formalist
approach) construes the numerical field as an operational field, on
the basis of certain singular axioms. This time, number occupies no
particular position as regards the laws of thought. It is a system of
rule-governed operations, specified in Peano's axioms by way of a
translucid notational practice, entirely transparent to the material
gaze. The space of numerical signs is simply the most 'originary' of
mathematics proper (preceded only by purely logical calculations).
We might say that here the concept of number is entirely mathema-
tised, in the sense that it is conceived as existing only in the course
of its usage: the essence of number is calculation.
The approach of Dedekind and Cantor, and then of Zermelo, von
Neumann and Godel (which we shall call the set-theoretical or 'pla-
tonising' approach) determines number as a particular case of the
hierarchy of sets. The fulcrum, absolutely antecedent to all construc-
tion, is the empty set; and 'at the other end', so to speak, nothing
prevents the examination of infinite numbers. The concept of number
is thus referred back to an ontology of the pure multiple, whose great
Ideas are the classical axioms of set theory. In this context, 'being a
number' is a particular predicate, the decision to consider as such
certain classes of sets (the ordinals, or the cardinals, or the elements
of the continuum, etc.) with certain distinctive properties. The essence
of number is that it is a pure multiple endowed with certain proper-
ties relating to its internal order. Number is, before being made
available for calculation (operations will be defined 'on' sets of
pre-existing numbers). Here we are dealing with an ontologisation of
1.4. My own approach will be as follows:
(a) The logicist perspective must be abandoned for reasons of inter-
nal consistency: it cannot satisfy the requirements of thought,
and especially of philosophical thought.
(b) The axiomatic, or operational, thesis is the thesis most 'prone'
to the ideological socialisation of number: it circumscribes the
question of a thinking of number as such within an ultimately
technical project.
(c) The set-theoretical thesis is the strongest. Even so, we must draw
far more radical consequences than those that have prevailed up
to the present. This book tries to follow the thread of these
conseq uences.
1.5. Whence my plan: To examine the theses of Frege, Dedekind
and Peano. To establish myself within the set-theoretical conception.
To radicalise it. To demonstrate (a most important point) that
in the framework of this radicalisation we will rediscover also
(but not only) 'our' familiar numbers: whole numbers, rational
numbers, real numbers, all, finally, thought outside of ordinary
operational manipulations, as subspecies of a unique concept of
number, itself statutorily inscribed within the ontology of the pure
1.6. Mathematics has already proposed this reinterpretation, as
might be expected, but only in a recessive corner of itself, blind to
the essence of its own thought: the theory of surreal numbers, invented
at the beginning of the 1970s by J. H. Conway (On Numbers and
Games, 1976),3 taken up firstly by D. E. Knuth (Surreal Numbers,
1974),4 and then by Harry Gonshor in his canonical book (An Intro-
duction to the Theory of Surreal Numbers, 1986).5 Any interest we
might have in the technical details of this theory will be here strictly
subordinated to the matter in hand: establishing a thinking of number
that, by fixing the latter's status as a form of the thinking of Being,
can free us from it sufficiently for an event, always trans-numeric, to
summon us, whether this event be political, artistic, scientific or
amorous. Limiting the glory of number to the important, but not
exclusive, glory of Being, and thereby demonstrating that what pro-
ceeds from an event in terms of truth-fidelity can never be, has never
been, counted.
1.7. None of the modern thinkers of number (I understand by this,
I repeat, those who, between Bolzano and Godel, tried to pin
down the idea of number at the juncture of philosophy and the
logico-mathematical) have been able to offer a unified concept
of number. Customarily we speak of 'number' with respect to
natural whole numbers,6 'relative' (positive and negative) whole
numbers, rational numbers (the 'fractions'), real numbers (those
that number the linear continuum) and, finally, complex numbers
and quaternions. We also speak of number in a more directly set-
theoretical sense when designating types of well-orderedness (the
ordinals) and pure quantities of any multiple whatsoever, including
infinite quantities (the cardinals). We might expect that a concept
of number would subsume all of these cases, or at least the more
'classical' among them, that is to say, the whole natural numbers (the
most obvious schema of discrete 'stepwise' enumeration) and the real
numbers (the schema of the continuum). But this is not at all
the case.
1.8. The Greeks clearly reserved the concept of number for whole
numbers, which was quite in keeping with their conception of the
composition of number on the basis of the One, since only natural
whole numbers can be represented as collections of units. To treat of
the continuum, they used geometrical denominations, such as the
relations between sizes or measurements. So their powerful concep-
tion was marked through and through by that division of mathemati-
cal disciplines on the basis of whether they treat of one or the other
of what were held by the Greeks to be the two possible types of
object: numbers (from which arithmetic proceeds) and figures (from
which, geometry). This division refers, it seems to me, to the two
orientations whose unity is dialectically effectuated by effective,
or materialist, thought: the algebraic orientation, which works by
composing, connecting, combining elements; and the topological
orientation, which works by perceiving proximities, contours and
approximations, and whose point of departure is not elementary
belongings but inclusion, the part, the subset.
This division is still
well-founded. Within the discipline of mathematics itself, the two
major divisions of Bourbaki's great treatise, once the general onto-
logical framework of set theory is set out, deal with 'algebraic struc-
tures' and 'topological structures'.
And the validity of this arrangement
subtends all dialectical thought.
1.9. It is nevertheless clear that, ever since the seventeenth century,
it has no longer been possible to place any sufficiently sophisticated
mathematical concept exclusively on one side of the opposition arith-
metic/geometry. The triple challenge of the infinite, of zero and of
the termination of the idea of the One disperses the idea of number,
shreds it into a refined dialectic of geometry and arithmetic, of the
topological and the algebraic. Cartesian analytic geometry radically
subverts the distinction from the very outset, and what we know
today as 'number-theory' had to call on the most complex resources
of 'geometry', in the extremely broad sense in which the latter has
been understood in recent decades. Moderns therefore can no longer
accept the concept of number as the object whose provenance is
foundational (the idea of the One) and whose domain is prescribed
(arithmetic). 'Number' is said in many senses. But which of these
senses constitutes a concept, allowing something singular to be
proposed to thought under this name?
1.10. The response to this question, in the work of the thinkers I
have mentioned, is altogether ambiguous and exhibits no unanimity
whatsoever. Dedekind, for example, can legitimately be named as the
first one to have, with the notion of the cut, convincingly 'generated'
the real numbers from the rationals.
But when he poses the question:
'What are numbers?' he responds with a general theory of ordinals
which certainly, as a particular case, might found the status of whole
numbers, but which cannot be applied directly to real numbers.1O In
which case, what gives us the right to say that real numbers are
'numbers'? Similarly, in The Foundations of Arithmetic" Frege offers
a penetrating critique of all previous definitions (including the Greek
definition of number as a 'set of units')'z and proposes a concept of
'cardinal number' that in effect subsumes - on the basis of certain
arguable premises, to which I shall later return - cardinals in the set-
theoretical sense, of which natural whole numbers represent the finite
case. But at the same time he excludes ordinals, to say nothing of
rational numbers, real numbers or complex numbers. To use one of
his favourite expressions, such numbers do not 'fall under the
[Fregean] concept' of number. Finally, it is clear that Peano's axiom-
atic defines whole numbers and them alone, as a rule-governed opera-
tional domain. Real numbers can certainly be defined directly with a
special axiomatic (that of a complete, totally ordered Archimedean
field). But, if the essence of 'number' resides in the specificity of the
statements constituting these axiomatics, then, given that these state-
ments are entirely dissimilar in the case of the axiomatic of whole
numbers and of that of real numbers, it would seem that, in respect
of their concept, whole numbers and real numbers have nothing
In common.
1.11. It is as if, challenged to propose a concept of number that can
endure the modern ordeal of the defection of the One, our thinkers
reserve the concept for one of its 'incarnations' (ordinal, cardinal,
whole, real ... ), without being able to account for the fact that the
idea and the word 'number' are used for all of these cases. More
particularly, they prove incapable of defining any unified approach,
any common ground, for discrete numeration (whole numbers), con-
tinuous numeration (real numbers) and 'general', or set-theoretical,
numeration (ordinals and cardinals). And yet it was precisely the
problem of the continuum, the dialectic of the discrete and the con-
tinuous, which, saturating and subverting the ancient opposition
between arithmetic and geometry, compelled the moderns to rethink
the idea of number. In this sense their work, admirable as it is in so
many ways, is a failure.
1.12. The anarchy thus engendered (and I cannot take this anarchy
to be innocent of the unthinking despotism of number) is so much
the greater in so far as the methods put to work in each case are
totally disparate:
(a) Natural whole numbers can be determined either by means of
a special axiomatic, at whose heart is the principle of recurrence
(Peano), or by means of a particular (finite) case of a theory of
ordinals, in which the principle of recurrence becomes a theorem
(b) To engender negative numbers, algebraic manipulations must
be introduced that do not bear on the 'being' of number, but
on its operational arrangement, on structures (symmetricisation
of addition).
(c) These manipulations can be repeated to obtain rational numbers
(symmetricisation of multiplication).
(d) Only a fundamental rupture, marked this time by a shift towards
the topological, can found the passage to real numbers (consid-
eration of infinite subsets of the set of rationals, cuts or Cauchy
seq uences).
(e) We return to algebra to construct the field of complex numbers
(algebraic closure of the Real Field, adjunction of the 'ideal'
element i = ~ or direct operational axiomatisation on pairs
of real numbers).
(f) Ordinals are introduced through the consideration of types of
order (Cantor), or through the use of the concept of transitivity
(von Neumann).
(g) The cardinals are treated through a totally different procedure,
that of biunivocal correspondence. 13
1.13. This arsenal of procedures was historically deployed according
to overlapping lines which passed from the Greeks, the Arab algebra-
ists and those of Renaissance Italy, through all the founders of modern
analysis, down to the 'structuralists' of modern algebra and the set-
theoretical creations of Dedekind and Cantor. How are we to extract
from it a clear and univocal idea of number, whether we think it as
a type of being or as an operational concept? All that the thinkers of
number have been able to do is to demonstrate the intellectual pro-
cedures that lead us to each species of 'number'. But, in doing so,
they left number as such in the shadow of its name. They remained
distant from that 'unique number which cannot be any other',I4
whose stellar insurrection Mallarme proposed.
1.14. The question, then, is as follows: is there a concept of number
capable of subsuming, under a single type of being answering to a
uniform procedure, at least natural whole numbers, rational numbers,
real numbers and ordinal numbers, whether finite or infinite? And
does it even make sense to speak of a number without at once speci-
fying which singular, irreducible apparatus it belongs to? The answer
is yes. This is precisely what is made possible by the marginal theory,
which I propose to make philosophically central, of 'surreal
This theory offers us the true contemporary concept of number,
and in doing so it overcomes the impasse of the thinking of number
in its modern-classical form, that of Dedekind, Frege and Cantor. On
its basis, and as the result of a long labour of thought, we can prevail
over the blind despotism of the numerical unthought.
1.15. We must speak not of a single age of the modern thinking of
number, but of what one might call, taking up an expression Natacha
Michel applies to literature, the 'first modernity' of the thinking of
number. IS The names of this first modernity are not those of Proust
and Joyce, but those of Bolzano, Frege, Cantor, Dedekind and Peano.
I am attempting the passage to a second modernity.
1.16. I have said that the three challenges to which a modern doc-
trine of number must address itself are those of the infinite, of zero
and of the absence of any grounding by the One. If we compare Frege
and Dedekind - so close on so many points - on this matter, we
immediately note that the order in which they arrange their responses
to these challenges differs in an essential respect:
On the infinite Dedekind, with admirable profundity, begins with the
infinite, which he determines with a celebrated positive property: 'A
system S is said to be infinite when it is similar to a proper part of
itself.'16 And he undertakes immediately to 'prove' that such an infi-
nite system exists. The finite will be determined only subsequently,
and it will be the finite that is the negation of the infinite (in which
regard Dedekind's numerical dialectic has something of the Hegelian
about it).17 Frege, on the other hand, begins with the finite, with
natural whole numbers, of which the infinite will be the 'prolonga-
tion' or the recollection in the concept.
On zero Dedekind abhors the void and its mark, and says so quite
explicitly: '[W]e intend here for certain reasons wholly to exclude the
empty system which contains no elements at all.'19 Whereas Frege
makes the statement 'zero is a number,20 the rock of his whole
On the One There is no trace of any privileging of the One in Frege
(precisely because he starts audaciously with zero). So one - rather
than the One - comes only in second place, as that which falls under
the concept 'identical to zero' (the one and only object that falls under
the concept being zero itself, we are entitled to say that the extension
of this concept is one). Dedekind, on the other hand, retains the idea
that we should 'begin' with one: 'the base-element 1 is called the
base-number of the number-series N'.21 And, correlatively, Dedekind
falls back without hesitation on the idea of an absolute AW
thought, an idea that could not appear as such in Frege's formalism:
'My own realm of thoughts, i.e. the totality S of all things, which can
be objects of my thought, is infinite.'23 Thus we see that, in retaining
the rights of the One, the All is supposed, because the All is that
which, necessarily, proceeds from the One, once the One is.
1.17. These divergences of order are no mere technical matter. They
relate, for each of these thinkers, to the respective centre of gravity
of their conception of number and - as we shall see - to the simulta-
neous stopping puint and founding point of their thought: the infinite
and existence for Dedekind, zero and the concept for Frege.
1.18. The passage to a second modernity of the thinking of number
obliges thought to return to zero, to the infinite and to the One. A
total dissipation of the One, an ontological decision as to the being
of the void and that which marks it, a lavishing without measure of
infinities: such are the parameters of such a passage. Unbinding from
the One delivers us to the unicity of the void and to the dissemination
of the infinite.
2.1. Frege
maintains that pure thought engenders number. Like
Mallarme, albeit without the effect of Chance, Frege thinks that
'every thought emits a dicethrow'.2 What is called Frege's 'logicism'
runs very deep: number is not a singular form of being, or a particular
property of things. It is neither empirical nor transcendent. Nor is it,
on the other hand, a constitutive category; it is deduced from the
concept. It is, in Frege's own words, a trait of the concept.
2.2. The pivotal property that permits the transition from pure
concept to number is that of a concept's extension. What does this
mean? Given any concept whatsoever, an object 'falls' under this
concept if it is a 'truth-case' of this concept, if the statement that
attributes to this object the property comprised in the concept is a
true statement. In other words, if the object satisfies the concept. Note
that everything originates with the truth-value of statements, which
is their denotation (truth or falsity). It could be said that, if the
concept generates number, it does so only in so far as there is truth.
Number is in this sense the index of truth, not an index of being.
But the idea of extension is ramified and obscure.
2.3. Given a concept, by the extension of that concept we mean all
the truth-cases (all objects qua truth-cases) that fall under this concept.
Every concept has an extension.
Now, take two concepts C
and C
• We will call them equinumer-
if there exists a biunivocal correspondence associating, object for
object, that which falls under concept C
with that which falls under
concept C
• That is, if a biunivocal correspondence can be defined
between the extension of C
and the extension of C
It is clear that Frege favours a 'cardinal' definition of number; that
he is not overly concerned with the structural order of that which falls
under the concept. And in fact this essential tool of biunivocity is
characteristic of all attempts to 'number' the multiple in itself, the pure
multiple subtracted from all structural considerations. To say that two
concepts are equinumerate is to say that they have the 'same quantity',
that their extensions are the same size, abstracting from any con-
sideration as to what the objects are that fall under those concepts.
2.4. Number consists in marking equinumeracy, the quantitative
identity of concepts. Whence the famous definition: 'The number
which belongs to the concept C is the extension of the concept "equi-
numerate to concept C".'5 Which means: every concept C generates
a number - namely, the set of concepts equinumerate to C, having
the 'same pure quantity', the same quantity of extension, as C. Note
that a number, grasped in its being, always designates a set of con-
cepts, namely all those that satisfy the statement 'is a concept equi-
numerate to C'.
2.5. The concept of number is constructed through the following
Concept Truth Objects that fall under the concept (that
satisfy the statement attributing the concept to the object)
Extension of the concept (all truth-cases of the concept) Equi-
numeracy of two concepts (via biunivocal correspondence of their
extensions) Concepts that fall under the concept of equinu-
me racy to a given concept C (that satisfy the statement 'is equinu-
merate to C') The extension of equinumeracy-to-C (the set of
concepts from the preceding stage) The number that belongs
to concept C (number is thus the name for the extension of
From a simplified and operational point of view, it could also
be said that, starting from the concept, we are able to pass through
the object on condition that there is truth; that we then compare
concepts, and that number names a set of concepts that have in
common a property made possible and defined by this comparison
(equinumeracy) .
2.6. To rediscover the 'ordinary', familiar numbers on the basis of
this pure conceptualism regulated by truth alone, Frege begins with
his admirable deduction of zero: zero is the number belonging to the
concept 'not identical to itself'.6 Since every object is identical to itself,
the extension of the concept 'not identical to itself' is empty. It
follows that zero is the set of concepts whose extension is empty and
which, by virtue of this, are equinumerate to the concept 'not
identical to itself'. Which is precisely to say that zero is that number
belonging to every concept whose extension is empty, is zero.
I have indicated in 1.17 the passage to the number 1: 'One' is the
number that belongs to the concept 'identical to zero,.7 It is interest-
ing to note that Frege emphasises, with regard to 1, that it has no
'intuitive' or empirical privilege, any more than it is a transcendent
foundation: 'The definition of 1 does not presuppose, for its objective
legitimacy, any matter of observed fact.,8 There can be no doubt that
Frege participates in the great modern process of the destitution of
the One.
The engendering of the sequence of numbers beyond 1 poses only
technical problems, which are resolved, in passing from n to n + 1,
by constructing between the extensions of corresponding concepts a
correlation such that the 'remainder' is exactly 1 - which has already
been defined.
2.7. Thus the deduction of number as a consequence of the concept
appears to have been accomplished. More exactly: from the triplet
conceptltruth!object, and from the single formal operator of biunivo-
cal correspondence, number emerges as an instance of pure thought,
or an integrally logical production; thought must presuppose itself,
in the form of a concept susceptible to having truth-cases (and there-
fore endowed with an extension). In so doing, thought presupposes
2.8. Why choose particularly the concept 'not identical to itself' to
ground zero? Any concept could be chosen so long as one is sure it
has an empty extension, that no thinkable object could have the
property it designates. For example 'square circle' - a concept which
in fact Frege declares is 'not so black as [it is] painted,.9 Since we
seek an entirely conceptual determination of number, the arbitrary
nature of this choice of concept is a little embarrassing. Frege is quite
aware of this, since he writes: 'I could have used for the definition of
nought any other concept under which no object falls."o But, to
obviate his own objection, he invokes Leibniz: the Principle of Iden-
tity, which says that every object is identical to itself, has the merit
of being 'purely logical'. 11 Purely logical? But we understood that it
was a matter of legitimating logico-mathematical categories (specifi-
cally, number) on the sole basis of the laws of pure thought. Isn't
there a risk of circularity if a logical rule is required right at the
outset? Now, equality is one of the logical, or operational, predicates
that require grounding (namely, equality between numbers). It might
be said, of course, that 'identical to itself' should not be confused
with 'equal to itself'. But if 'identity' must here indeed be carefully
distinguished from the logical predicate of equality, it is nevertheless
equally clear that the statement 'every object is identical to itself' is
not a 'purely logical' statement. It is an onto-logical statement. And,
qua ontological statement, it is immediately disputable: no Hegelian,
for example, would admit the universal validity of the principle of
identity. For our hypothetical Hegelian, the extension of the concept
'not identical to itself' is anything but empty!
2.9. The purely a priori determination of a concept certain to have
an empty extension is an impossible task without powerful prior
ontological axioms. The impasse that Frege meets here is that of an
unchecked doctrine of the object. For, from the point of view of the
pure concept, what is an 'object' in general, any object whatsoever,
taken from the total Universe of objects? And why is the object
required to be identical to itself, when the concept is not even required
to be non-contradictory in order to be legitimate, as indicated
by Frege's positive regard for concepts of the 'square circle' type,
which, he stresses, are concepts like any other? Why would the
law of the being of objects be more stringent than the law of the
being of concepts? Doubtless it would be so if one were to accept
Leibnizian ontology, for which existent objects obey an other
principle than do thinkable objects, the Principle of Sufficient
Reason. It thus appears that the deduction of number on the basis
of the concept is not so much universal, or 'purely logical', as it is
2.10. To posit as self-evident that the extension of a concept is this
or that (for example, that the extension of the concept 'not identical
to itself' is empty) is tantamount to supposing that we can move
unproblematically from concept to existence, given that the extension
of a concept brings into play the 'objects' that fall under this concept.
A generalised ontological argument is at work here, and it is this very
argument that subtends the deduction of number on the basis of the
concept alone: number belongs to the concept through the mediation
of the thinkable objects that fall under the concept.
2.11. The principal thought-content of Russell's paradox, com-
municated to Frege in 1903, is its undermining of every pretension
to legislate over existence on the basis of the concept alone, and
especially over the existence of the extension of concepts. Russell
presents a concept (in Frege's sense) - the concept 'to be a set that is
not an element of itself' - which is surely a wholly proper concept
(more so, truth be told, than 'not identical to itself'), but one none-
theless whose extension does not exist. It is actually contradictory to
suppose that 'objects' - in this instance, sets - that 'fall under this
concept' themselves form a set.
And, if they 40 not form a set, then
no biunivocal correspondence whatsoever can be defined for them.
So this 'extension' does not sustain equinumeracy, and consequently
no number belongs to the concept 'set that is not an element of
itself' .
The advent, to the concept, of an innumerable ruins Frege's
general deduction. And, taking into account the fact that the para-
doxical concept in question is quite ordinary (for example, the concept
is valid for all the sets customarily used by mathematicians: they are
not elements of themselves), we might well suspect that there exist
many other concepts to which no number belongs. In fact, it is impos-
sible to predict a priori the extent of the disaster. Even the concept
'not identical to itself' could well turn out not to have any existent
extension, which is something entirely different from having an empty
extension. Let's add that Russell's paradox is purely logical, that is
to say, it is precisely proven: to admit the existence of a set of all
those sets that do not belong to themselves undermines deductive
language by introducing a formal contradiction (the equivalence
between a proposition and its negation).
2.12. A sort of 'repair' was proposed by Zermelo.
It consists in
saying that we can conclude from the concept the existence of its
extension on condition that we operate within an already given exis-
tence. Given a concept C and a domain of existing objects, we can
say that there exists, in this existing domain, the set of objects that
fall under this concept - i.e. the extension of the concept. Obviously,
this extension is relative to a domain specified in advance and does
not exist 'in itself'. This is a major ontological transformation: within
this new framework it is not possible to move from concept to exis-
tence (and thus to number); we can only move to an existence that
is somehow carved out of a pre-given existence. We can 'separate' in
a given domain those objects within it that validate the property
exposed by the concept. This is why Zermelo's principle, which dras-
tically limits the rights of the concept and of language over existence,
is called the Axiom of Separation. And it does indeed seem that
accepting this axiom safeguards us against the inconsistency-effects
of Russell-type paradoxes.
2.13. Russell's paradox is not paradoxical in the slightest. It is a
materialist argument, because it demonstrates that multiple-being is
anterior to the statements that affect it. It is impossible, says the
'paradox', to accord to language and to the concept the right of
unfettered legislation over existence. Even supposing that there is a
transcendental function of language, it supposes also the availability
of some prior existent, the power of this function being simply that
of carving out or delimiting extensions of the concept within this
specified existent.
2.14. Can we, in assuming Zermelo's axiom, save the Fregean con-
struction of number? Once again, everything turns on the question
of zero. I might proceed in the following way: given a delimited
domain of objects, whose existence is somehow externally guaran-
teed, I will call 'zero' (or 'empty set', which is the same thing) that
which detaches, or separates, within this domain, the concept 'not
identical to itself', or any other such concept under which I can assure
myself that no objects of the domain fall. As we are dealing with a
limited domain, and not, as in Frege's construction, with 'all objects'
(a formulation that led to the impasse of a Leibnizian choice without
criteria), there is a chance of my finding such a concept. If, for
example, I take a set of black objects, I will call 'zero' that which
separates in this set the concept 'to be white'. The rest of the con-
struction will follow.
2.15. But what domain of objects could I start with, for which it can
be guaranteed that these objects pertain to pure thought, that they
are 'purely logical'? Recall that Frege intends to construct a concept
of number that is, according to his own expression, 'not ... either
anything sensible or a property of an external thing', 14 and that he
emphasises on several occasions that number is subtracted from the
representable. Establishing that number is a production of thought,
deducing it from the abstract attributes of the concept in general -
this cannot be achieved using black and white ohjects. The question
then becomes: what existent can I assure myself of, outside of any
experience? Is the axiom 'something exists' an axiom of pure thought
and, supposing that it is, can I discern any property of which it
is certain that it does not belong in any way to this existent
2.16. A 'purely logical' demonstration of existence, for thought, of
a nondescript object, a point of being, an 'object = x': the statement
'every x is equal to itself' is an axiom of logic with equality. Now,
the universal rules of first-order logic, a logic valid for every domain
of objects, allow us to deduce, from the statement 'every x is equal
to x', the statement 'there exists an x that is equal to x' (subordina-
tion of the existential quantifier to the universal quantifier).15 There-
fore, there exists x (at least that x which is equal to itself).
Thus we can demonstrate within the framework of set theory, first
of all, by purely logical means, that a set exists. Then we can separate
the empty set within that existent whose existence has been proved,
by utilising a property that no element can satisfy (for example, 'is
not equal to itself'). We have respected Zermelo's axiom, since we
have operated within a prior existent, but we have succeeded in
engendering zero.
2.17. It is quite obvious, I think, that this 'proof' is an unconvincing
artifice, a logical sleight of hand. From the universal postulate of
self-equality (which we might possibly accept as an abstract law, or
a law of the concept), who could reasonably infer that there exists
something rather than nothing? If the universe were absolutely void,
it would remain logically admissible that, supposing that something
existed (which would not be the case), it would have to be equal to
itself. The statement 'every x is equal to x' would be valid, but there
would be no x, so the statement 'there exists an x equal to itself'
would not be valid.
The passage from universal statement to assertion of existence is
an exorbitant right, which the concept cannot arrogate to itself. It is
not possible to elicit existence on the basis of a universal law that
could be upheld just as well in absolute nothingness (consider for
example the statement 'the nothing is identical to itself'). And, since
no existent object can be deduced from pure thought, you cannot
separate zero therein. Zermelo does not save Frege.
2.18. The existence of zero, or of the empty set, and therefore the
existence of numbers, is in no way deducible from the concept, or
from language. 'Zero exists' is inevitably a first assertion; the very
one that fixes an existence from which all others will proceed. Far
from it being the case that Zermelo's axiom, combined with Frege's
logicism, allows us to engender zero and then the chain of numbers,
it is on the contrary the absolutely inaugural existence of zero (as
empty set) that ensures the possibility of separating any extension of
a concept whatsoever. Number comes first here: it is that point of
being upon which the exercise of the concept depends. Number, as
number of nothing, or zero, sutures every text to its latent being. The
void is not a production of thought, because it is from its existence
that thought proceeds, in as much as 'it is the same thing to think
and to be'.16 In this sense, it is the concept that comes from number,
and not the other way around.
2.19. Frege's attempt is unique in certain regards: it is not a matter
of creating new intra-mathematical concepts (as will be the case in
Dedekind and Cantor), but of elucidating - with the sole resource of
rigorous analysis - what, among the possible objects of thought,
singularises those which fall under the concept of number. In this
respect, my own efforts follow along the same lines. We simply need
to remove the obstacles by reframing the investigation according to
new parameters. Above all, it must be shown that thought is not
constituted by concepts and statements alone, but also by decisions
that engage it within the epoch of its exercise.
Additional Note on a
Contemporary Usage of Frege
- -
3.1. Jacques-Alain Miller, in a 1965 lecture entitled 'Suture' and
subtitled 'Elements of the logic of the signifier', 1 put forward a reprise
of Frege's construction of number. His text founds a certain regime
of compatibility between structuralism and the Lacanian theory of
the subject. 1 am myself periodically brought back to this foundation,2
albeit only on condition of disrupting it somewhat. Twenty-five years
later, 'I am here; 1 am still here'.3
3.2. Miller puts the following question to Frege: 'What is it that
functions in the sequence of whole numbers?,4 And the response to
this question - a response, might 1 say, forcefully extorted out of
Frege - is that 'in the process of the constitution of the sequence, the
function of the sub;ect, unrecognised,s is operative'.
3.3. If we take this response seriously, it means that, in the last
instance, in the proper mode of its miscognition, it is the function of
that subject whose concept Lacan's teaching communicates to us that
constitutes, if not the essence, at least the process of engenderment
(the 'genesis of progression', says Millerf of number.
Obviously such a radical thesis cannot be ignored. Radical, it
would seem at first glance, with regard to Frege's doctrine, which
dedicates a specific argument to the refutation of the idea that number
might be 'subjective,8 (although it is true that, for Frege, 'subjective'
means 'caught up in representation', which obviously does not match
the Lacanian function of the subject), Radical also with regard to my
own thesis, since I hold that number is a form of being, and that, far
from being subtended by the function of the subject, it is on the
contrary on the basis of number, and especially of that first number-
being that is the void (or zero), that the function of the subject
receives its small share of being.
3.4. We will not undertake here to examine the importance of this
text - the first great Lacanian text not to be written by Lacan himself
- for the doctrine of the signifier, nor to explore what analogy it
employs to illuminate the importance - at the time, still little appreci-
ated - of all that the master taught us as to the subject's being com-
prised in the effects of a chain. We seek to examine exclusively what
Miller's text assumes and proposes with regard to the thinking of
number as such.
3.5. Miller's demonstration is organised as follows:
• To found zero, Frege (as we saw in 2.6) summons to his aid the
concept 'not identical to itself'. No object falls under this concept.
On this point, Miller emphasises - even compounds - Frege's refer-
ence to Leibniz. To suppose that an object could be not be identical
to itself, or that it could be non-substitutable for itself, would be
entirely to subvert truth. In order to be true, a statement bearing upon
object A must suppose the in variance of A in each occurrence of the
statement, 'each time' the statement is made. The principle 'A is A'
is a law of any possible truth. And reciprocally, in order that truth
be saved, it is crucial that no object should fall under the concept
'not identical to itself'. Whence zero, which numbers the extension
of such a concept.
• Number is thus shown to issue from the concept alone, on condi-
tion of truth. But this demonstration is consistent only because it has
been able to invoke in thought an object non-identical to itself, even
if only to discharge it in the inscription of zero. Thus, Miller writes,
'the 0 which is inscribed in the place of the number consummates the
exclusion of this object'.
To say that 'no object' falls under the concept 'not identical to
itself' is to make this object vanish as soon as it is invoked, in this
nothing the only subsisting trace of which will be, precisely, the mark
zero: 'Our purpose has been,' Miller concludes, 'to recognize in the
zero number the suturing stand-in for the lack'. 10
• What is it that comes to lack thus? What 'object' can have as a
stand-in for its own absence the first numerical mark; and support,
in relation to the whule chain of numbers, the uninscribable place of
that which appears only in order to vanish? What is it that insists
between numbers? We must certainly agree that no 'object' can, even
by failure or default, fall in that empty place that assigns non-
self-identity, But there does exist (or here, more precisely, ek-sist)
precisely that which is not object, that which is proper to the non-
object, the object as impossibility of the object: the subject. 'The
impossible object, which the discourse of logic summons as the not-
identical with itself and then rejects. , . in order to constitute itself as
what it is, which it summons and rejects wanting to know nothing
of it, we name this object, in so far as it functions as the excess which
operates in the sequence of numbers, the subject.' II
3.6. We must meticulously distinguish between that which Miller
assumes from Frege and that which he deciphers in Frege's work on
his own account, I will proceed in three stages.
3.7. FIRST STAGE Miller takes as his starting point the proposition
of Leibniz-Frege according to which salva veritate
demands that all
objects should be identical to themselves. The whole literalisation of
the real towards which Leibniz worked all his life, and to which
Frege's ideography is the undoubted heir, is in fact surreptitiously
assumed here. In this regard, Miller is indeed right to equate, along
with Leibniz, 'identical to itself' and 'substitutable', thus denoting an
equivalence between the object and the letter. For what could it mean
to speak of the substitutability of an object? Only the letter is entirely
substitutable for itself. 'A is A' is a principle of letters, not of objects.
To be identifiable at a remove from itself, and subject to questions
of substitutability, the object must fall under the authority of the
letter,13 which alone renders it over to calculation. If A is not identical
at all moments to A, truth (or rather veridicality) as calculation
The latent hypothesis is therefore that truth is of the order of cal-
culation. It is only on this supposition that, firstly, the object has
to be represented as a letter; and, secondly, that the non-self-identity
of the object-letter radically subverts truth. And if truth is of the
order of calculation, then zero - which numbers the exclusion
of the non-self-identical (the subject) - is itself nothing but a letter,
the letter O. The conclusion then follows straightforwardly that
zero is the inert stand-in for lack, and that what 'drives' the sequence
of numbers as a product of marks - a repetition in which is articu-
lated the miscognition of that which insists - is the function of the
More simply: if truth is saved only by upholding the principle of
identity, then the object emerges in the field of truth only as a letter
amenable to calculation. And, if this is the case, number can sustain
itself only as the repetition of that which insists in lacking, which is
necessarily the non-object (or the non-letter, which is the same thing),
the place where 'nothing can be written'\4 - in short, the subject.
3.8. No one is obliged to be a Leibnizian, even if we must recognise
in this philosophy the archetype of one of the three great orientations
in thought, the constructivist or nominalist orientation (the other two
being the transcendent and the generic).\5 As an advocate of the
generic orientation, I declare that, for truth to be saved, one must
precisely abolish those two great maxims of Leibnizian thought, the
Principle of Non-Contradiction and the Principle of Indiscernibles.
3.9. A truth supposes that the situation of which it is the truth attains
non-self-identity: this non-self-identity is indicated by the situation's
being supplemented by an 'extra' multiple, one whose belonging or
non-belonging to the situation is, however, intrinsically undecidable.
I have named this supplement 'event', and it is always from an event
that a truth-process originates. Now, when the undecidable event
must be decided within the situation, that situation necessarily under-
goes a vacillation as to its identity.
3.10. The process of a truth - puncturing the strata of knowledge
harboured by the situation - inscribes itself within the situation as
indiscernible infinity, which no thesaurus of established language has
the power to designate.
Let's say simply that zero, or the void, has nothing in itself to do
with the salvation of truth, which is at play in the 'laboured' correla-
tion between the undecidability of the event and the indiscernibility
of its result within the situation. No more so than it is possible to
refer truth to the power of the letter, since the existence of a truth is
precisely that to which no inscription can attest. The statement 'truth
is' - far from guaranteeing that no object falls under the concept of
'not identical to itself' and that therefore zero is the number of that
concept - instead allows us this threefold conclusion:
- there exists an object that has attained 'non-self-identity' (unde-
cidability of the event);
- there exist an infinity of objects that do not fall under any concept
(indiscernibility of a truth);
- number is not a category of truth.
3.11. SECOND STAGE What is the strategy of Miller's text? And
what role does number as such play within it? Is it really about
arguing that the function of the subject is implicated - as a miscog-
nised foundation - in the essence of number? This is undoubtedly
what is stated in all clarity by the formula I have already cited above:
'In the process of the constitution of the sequence [of numbers 1 ... the
function of the subject ... is operative,'16 More precisely, only the
function of the subject - that which zero, as number, marks in the
place of lack, holding the place of its revocation - is capable of
explaining what, in the sequence of numbers, functions as iteration
or repetition: being excluded, the subject (the non-self-identical)
includes itself through the very insistence of marks, incessantly repeat-
ing the 'one more step', firstly from 0 to 1 ('the 0 counts for 1', notes
Miller), then indefinitely, from n to n + 1: 'its [the subject's - in the
Lacanian sense 1 exclusion from the field of number is identified with
repetition'. 17
3.12. Other passages of Miller's text are more equivocal, indicating
an analogical reading. For example: 'If the sequence of numbers,
metonymy of the zero, begins with its metaphor, if the zero member
of the sequence as number is only the stand-in suturing the absence
(of the absolute zero) which moves beneath the chain according to
the alternation of a representation and an exclusion - then what is
there to stop us from recognising in the restored relation of the zero
to the sequence of numbers the most elementary articulation of the
subject's relation to the signifying chain?,18 The word 'recognising' is
compatible with the idea that the Fregean doctrine of number pro-
poses a 'matrix' (the title of another article by Miller on the same
question)19 that is isomorphic with (maximum case) or similar to
(minimum case), but in any case not identical to, the relation of the
subject to the signifying chain. Frege's doctrine would then be a per-
tinent analogon of Lacanian logic: to which we would have no reply,
since in that case Miller's text would not be a text about number. It
would be doubly not so: firstly because it would speak, not of number,
but of Frege's doctrine of number (without taking any position on
the validity or consistency of that doctrine); and secondly because it
would present the sequence of numbers as a didactic vector for the
logic of the signifier, and not as an effective example of an implica-
tion of the function of the subject in the sequence of numbers.
3.13. This critical evasion assumes that two conditions are met: that
there is, between the doctrine of number and that of the signifier,
isomorphism or similarity, and not identity or exemplification; and
that Miller does not account for the validity of the Fregean doctrine
of number.
3.14. On this last point, where, to my mind (that is, to one who is
concerned with the thinking of number as such), everything hangs in
the balance, Miller maintains the suspense at every step. He speaks
of 'Frege's system' without our being able to decide whether or not,
in his opinion, the latter is an actually accomplished theory of number,
a theory entirely defensible in essence. It is striking that at no point
in this very subtle and intricate exercise are the immanent problems
of 'Frege's system' ever raised - in particular, those that I highlighted
above with regard to zero, the impact of Russell's paradox, Zermelo's
axiom and, ultimately, the relation between language and existence.
It thus remains possible to believe that the isomorphism signifier/
number operates between, on the one hand, Lacan and, on the other,
Frege reduced to a singular theory whose inconsistency is of no con-
sequence with regard to the analogical goals pursued.
3.15. Evidently, it remains to be seen whether this inconsistency isn't,
as a result, transferred to the other pole of the analogy, that is, to
the logic of the signifier. The risk is not inconsequential, given that
Miller places the latter in a founding position with regard to logic
tout court - presumably including Frege's doctrine: 'The first [the
logic of the signifier] treats of the emergence of the second [the logic
of logicians], and should be conceived of as the logic of the origin of
logic.'20 But what happens if the completion of this process of origi-
nation is induced, through the theme of the subject, by a scheme
(Frege's) marred by inconsistency? But this is not my problem. Given
the conditions I have laid out, if the text is not about number, then
we are finished here.
3.16. THIRD STAGE There remains, however, an incontestable
degree of adherence on Miller's part to a common representation of
number, wherein number is conceived of as in some way intuitive,
and which I cannot accept. This concerns the idea - central, since it
is precisely here that the subject makes itself known as the cause of
repetition - according to which number is grasped as a 'functioning',
or in the 'genesis of the progression'. This is the image of a number
that is 'constructed' iteratively, on the basis of that point of puncture
that is denoted by zero. This dynamical theme, which would have us
see number as passage, as self-production, as engenderment, is omni-
present in Miller's text. The analysis centres precisely on the 'passage'
from 0 to 1, or on the 'paradox of engendering' n + 1 from n.
3.17. This image of number as iteration and passage precludes any
orderly discussion of the essence of number. Even if we can only
traverse the numeric domain according to certain laws of progression,
of which succession is the most common (but not the only one, far
from it), why must it follow that these laws are constitutive of the
being of number? It is easy to see why we have to 'pass' from one
number to the next, or from a sequence of numbers to its limit. But
it is, to say the least, imprudent thereby to conclude that number is
defined or constituted by such passage. It might well be (and this is
my thesis) that number does not pass, that it is immemorially deployed
in a swarming
coextensive with its being. And we will see that, just
as these laborious passages only govern our passage through this
deployment, in the same way it is likely that we remain ignorant of,
have at the present time no use for, or no access to, the greater part
of those numbers that our thought can conceive of as existent.
3.18. The 'constructivist' thesis, which makes of iteration, succes-
sion, passage the very essence of number, leads to the conclusion that
very few numbers exist, since here 'exist' has no sense apart from
that effectively supported by some such passage. Certainly, intuition-
ists assume this impoverished perspective. Even a semi-intuitionist
like Borel
thought that the great majority of natural whole numbers
'don't exist' except as a fictional and inaccessible mass. So it might
well be that the Leibnizian choice that Miller borrows from Frege is
doubled by a latent intuitionist choice.
We must recognise that intuitionist logic and the logic of the signi-
fier have more than a little in common, if only because the former
expressly invokes the subject (the 'mathematician-subject') as part of
its machinery. But in my opinion such a choice would represent an
additional reason not to enter into a doctrine of number whose
overall effect is to make the place of number, measured by the opera-
tional intuition of a subject, inexorably finite. For the domain of
number is rather an ontological prescription incommensurable with
any subject and immersed in the infinity of infinities.
3.19. The problem now becomes: how to think number whilst admit-
ting, against Leibniz, that there are real indiscernibles; against the
intuitionists, that number persists and does not pass; and against the
foundational use of the subjective theme, that number exceeds all
4.1. Dedekind I introduces his concept of number within the frame-
work of what we would today call a 'na'ive' theory of sets. 'Na'ive'
because a theory of multiplicities is advanced that recapitulates
various presuppositions about things and about thought. 'Na'ive'
meaning, in fact: philosophical.
Dedekind states explicitly, in the opening of his text The Nature
and Meaning of Numbers, that he understands 'by thing every object
of our thought';2 and, a little later, that, when different things are
'for some reason considered from a common point of view, associated
in the mind, we say that they form a system S'.3 A system in Dede-
kind's sense is therefore quite simply a set in Cantor's sense. The
space of Dedekind's work is not the concept (as in Frege), but,
directly, the pure multiple, a collection that counts for one (as a
system) objects of thought.
4.2. Dedekind develops a conception of number that (like Cantor's)
is essentially ordinal. We saw (compare 2.3) that Frege's conception
was essentially cardinal (proceeding via biunivocal correspondences
between extensions of concepts). What is the significance of this dis-
tinction? In the ordinal view, number is thought as a link in a chain,
it is an element of a total order. In the cardinal view, it is rather the
mark of a 'pure quantity' obtained through the abstraction of domains
of objects having 'the same quantity'. The ordinal number is thought
according to the schema of a sequence, the cardinal number, accord-
ing to that of a measurement.
4.3. Dedekind affirms that infinite number (the totality of whole
numbers, for example) precedes, in construction, finite number (each
whole number, its successor, and so on). Thus the existence of an
infinite (indeterminate) system, and then the particular existence of
N (the set of natural whole numbers) form the contents of the para-
graphs numbered 66 and 72 in Dedekind's text, whereas a result as
apparently elementary as 'every number n is different from the fol-
lowing number n" comes only in paragraph 81.
Dedekind is a true modern. He knows that the infinite is simpler
than the finite, that it is the most general attribute of being, an intu-
ition from which Pascal had already drawn radical consequences -
and was the first to do so - as to the site of the subject.
4.4. Dedekind first of all invites us to accept the philosophical concept
of 'system', or any multiplicity whatsoever (compare 4.1). The prin-
cipal operator will then be, as in Frege (2.3), the idea of biunivocal
correspondence between two systems. Dedekind, however, will make
use of it in a totally different way than did Frege.
Let's note in passing that the biunivocal correspondence, bijection,
is the key notion of all the thinkers of number of this epoch. It orga-
nises Frege's thought, Cantor's and Dedekind's.
4.5. Dedekind calls the function, or correspondence, a 'transforma-
tion',4 and what we would call a bijective function or a biunivocal
correspondence he calls a 'similar transformation'.s In any case, we
are dealing with a function (which makes every element of a set (or
system) S' correspond to an element (and one only) of a set S, in such
a way that:
- to two different elements 51 and 52 of S will correspond two dif-
ferent elements ((sd and ((52) of S/;
- every element of S' is the correspondent, through (, of an element
of S.
A distinct (today we would say injective) function is a function
that complies only with the first condition:
Evidently, such functions can be defined 'in' a system S, rather
than 'between' a system S and another system S/. Functions (or
transformations) of this type make every element of S correspond
to an element of S (either another element or the same one: the
function could be the function of identity, at least for the element in
4.6. Take, then, a system S, an application ( (not necessarily one of
likeness or a biunivocal one) of S to itself, and s, an element of S. We
will call the chain of the element s for the application (, the set of
values of the function obtained by iterating it starting from s. So the
chain of s for (is the set whose elements are: s,((s),((((s)), ((((((s))), ... ,
We are not necessarily dealing here with an infinite iteration: it
could very well be that, at a certain stage, the values thus obtained
would repeat themselves. This is evidently the case if S is finite, since
the possible values, which are the elements of S (the application (
operates from S within S), will be exhausted after a finite number of
stages. But it would also be the case were one to come across a value
p of the function (where, for p, (is identical. Because then ((p) = p,
and therefore ((((p)) = ((p) = p. The function halts at p.
4.7. We will say that a system N is (this is Dedekind's expression)
simply in{inite
if there exists a transformation ( of N within N that
complies with the three following conditions:
1 The application ( of N within N is a distinct application (d.
2 N is the chain of one of its elements, which Dedekind denotes by
1, and which he calls the base-element of N.
3 The base-element 1 is not the correspondent through ( of any
element of N. In other words, for any n which is part of N,
f(n) ...;:. 1: the function (never 'returns' to 1.
We can form a simple enough image of such an N. We 'start' with
the element 1. We know (condition 3) that ((1) is an element of N
different from 1. Next we see that ((((1)) is different from 1 (which
is never a value for f). But, equally, ((((1)) is different from ((1). In
fact, the function ((condition 1) is a distinct transformation - so two
different elements must correspond, through (, to different elements.
From the fact that 1 is different from (( 1) it follows that (( 1) is dif-
ferent from ((((1)). More generally, every element obtained through
the iteration of function (will be different from all those that 'pre-
ceded' it. And, since N (condition 2) is nothing other than the chain
thus formed, N will be composed of an 'infinity' (in the intuitive
sense) of elements, all different, ordered by function (, in the sense
that each element 'appears' through an additional step of the
process that begins with 1 and is continued by repeatedly applying
operation f.
4.8. The 'system' N thus defined is the place of number. Why?
Because all the usual 'numerical' manipulations can be defined on the
elements n of such a set N.
By virtue of the function f, we can pass without difficulty to the
concept of the 'successor' of a number: if n is a number, fIn) is its
successor. It is here that Dedekind's 'ordinal' orientation comes into
effect: function f, via the mediation of the concept of the chain, is
that which defines N as the space of a total order. The first 'point'
of this order is obviously 1. For philosophical reasons (compare
1.17), Dedekind prefers a denotation beginning with 1 to one begin-
ning with 0; '1' denotes the first link of a chain, whereas zero is
'cardinal' in its very being: it marks lack, the class of all empty
With 1 and the operation of succession it will be easy to obtain,
firstly, the primitive theorems concerning the structure of the
order of numbers, and then the definition of arithmetical operations,
addition and multiplication. On the sole basis of the concepts
of 'system' (or set) and of 'similar transformation' (or biunivocal
correspondence), the 'natural' kingdom of numericality will be
4.9. A system N, structured by a function f which complies with the
three conditions above (4.7) will be called 'a system of numbers', a
place of the set of numbers. To cite Dedekind:
If, in the consideration of a simply infinite system N, set in order by
a transformation t, we entirely neglect the special character of the ele-
ments, simply retaining their distinguishability and taking into account
only the relations to one another in which they are placed by the order-
setting transformation t, then are these elements called natura/numbers
or ordinal numbers or simply numbers, and the base-element 1 is called
the base-number of the number-series N. With reference to this freeing
the elements from every other content (abstraction), we are justified in
calling numbers a free creation of the human mind.
The enthusiastic tone leaves no room for doubt: Dedekind is con-
scious of having, with his purely functional and ordinal engendering
of 'system'S, torn number away from any form of external jurisdic-
tion, in the direction of pure thought. This was already the tone, and
these the stakes, of the 'proclamation' that appeared in the Preface
to the first edition of his pamphlet: 'In speaking of arithmetic (algebra,
analysis) as a part of logic, I mean to imply that I consider the number
concept to be entirely independent of the notions or intuitions of
space and time, that I consider it more as an immediate result of the
laws of thought.' This is a text that, as will be appreciated, lends
itself to a Kantian interpretation: the whole problem for modern
thinkers of number is to navigate within the triangle Plato-Kant-
In defining, not 'a' number, but N, the simply infinite
'system' of numbers, Dedekind considers, with legitimate pride, that
he has established himself, by means of the power of thought alone,
in the intelligible place of numericality.
4.10. Informed by Frege's difficulties, which do not concern his
concept of zero and of number, but the transition from concept to
existence or the jurisdiction of language over being, we ask: does a
system of numbers, a 'simply infinite' system N, exist? Or will some
unsuspected 'paradoxes' come to temper, for us, Dedekind's intel-
lectual enthusiasm?
4.11. Dedekind is evidently concerned about the existence of his
system of number. In order to establish it, he proceeds in three
1 Intrinsic definition, with no recourse to philosophy or to intuition,
of what an infinite system (or set) is.
2 Demonstration (this, as we shall see, highly speculative) of the
existence of an infinite system.
3 Demonstration of the fact that all infinite systems 'contain as
a proper part a simply infinite system N'.
These three steps permit the following conclusion to be drawn:
since at least one infinite system exists, and every infinite system has
as a subsystem an N - a simply infinite system or 'place of number'
- then this place exists. Which is to say: number exists. The idea that
'arithmetic should be a part of logic,9 signifies that, by means of the
conceptual work of pure thought alone, I can guarantee the consis-
tency of an intelligible place of numericality, and the effective exis-
tence of such a place.
4.12. Dedekind's definition of an infinite set is remarkable. He
himself was very proud of it, and with good reason. He notes that
'the definition of the infinite ... forms the core of my whole investiga-
tion. All other attempts that have come to my knowledge to distin-
guish the infinite from the finite seem to me to have met with so little
success that I think I may be permitted to forgo any critique of
This definition of the infinite systematises a remark already made
by Galileo: there is a biunivocal correspondence between the whole
numbers and the numbers that are their squares. Suffice to say, f(n)
= n
• However, the square numbers constitute a proper part of the
whole numbers (a proper part of a set is what we call a part that is
different from the whole, a truly 'partial' part). It seems, therefore,
in examining intuitively infinite sets, that there exist biunivocal cor-
respondences between the sets as a whole and one of their proper
parts. This part, then, has 'as many' elements as the set itself. Galileo
concluded that it was absurd to try to conceive of actual infinite sets.
Since an infinite set is 'as large' (contains 'as many' elements) as one
of its proper parts, the statement 'the whole is greater than the part'
is apparently false in the case of infinite totalities. Now, this statement
is an axiom of Euclid's Elements, and Galileo did not think it could
be renounced.
Dedekind audaciously transforms this paradox into the definition
of infinite sets: 'A system S is said to be infinite when it is similar to
a proper part of itself. In the contrar} case, S is said to be a finite
system,'ll (Remember that, in Dedekind's terminology, 'system'
means set, and the similarity of two systems means that a biunivocal
correspondence exists between them).
4.13. The most striking aspect of .Dedekind's definition is that it
determines infinity positively, and subordinates the finite negatively.
This is its especially modern accent, such as is almost always found
in Dedekind. An infinite system has a property of an existential
nature: there exists a biunivocal correspondence between it and one
of its proper parts. The finite is that for which such a property does
not obtain. The finite is simply that which is not infinite, and all the
positive simplicity of thought hinges on the infinite. This intrepid
total secularisation of the infinite is a gesture whose virtues we (inept
partisans of 'finitude', wherein our religious dependence can still be
read) have not yet exhausted.
4.14. The third point of Dedekind's approach (that every infinite
system contains as one of its parts a system of type N, a place of
number, see 4.11) is a perfectly elegant proof.
Suppose that a system S is infinite. Then, given the definition of
infinite systems, there exists a biunivocal correspondence f between
S and one of its proper parts S'. In other words a bijective function
f that makes every element of 5 correspond to an element of 5'. Since
5' is a proper part of 5, there is at least one element of 5 that is not
in the part 5' (otherwise 5 = 5', and 5' is not a 'proper' part). We
choose such an element, and call it 1. Consider the chain of 1 for the
function f (for 'chain' d. 4.6). We know that:
• f is a distinct (injective) transformation, or function, since it is
precisely the biunivocal correspondence between 5 and 5', and all
biunivocal correspondence is distinct;
• 1 certainly does not correspond through f to any other term of
the chain, since we have chosen 1 from outside of 5', and f only
makes elements of S' correspond to elements of 5. An element s
such that f(s) = 1 therefore cannot exist in the chain. In the chain,
the function never 'returns' to 1.
The chain of 1 for fin 5 is, then, a simply infinite set N: it complies
with the three conditions set for such an N in 4.7 above.
We are thereby assured that, if an infinite system 5 exists, then an
N, a place of number, also exists as part of that 5. Dedekind's thesis
is ultimately as follows: if the infinite exists, number exists. This point
(taking account of the ordinal definition of number as the chain of
1 for a similar transformation, and of the definition of the infinite)
is exactly demonstrated.
4.15. But does the infinite exist? There lies the whole question. This
is point two of Dedekind's approach, where we see that, for Dede-
kind, the infinite, upon which the existence of number depends,
occupies the place which for Frege is occupied by zero.
4.16. To construct the proof upon which henceforth all will rest (the
consistency and the existence of an infinite system or set), Dedekind
briskly canvasses all his initial philosophical presuppositions (the
thing as object of thought). Of course, these presuppositions already
quietly prop up the very idea of a 'system' (collection of anything
whatsoever). But, seized by the superbly smooth surface of the sub-
sequent definitions (chain, simply infinite set) and proofs, we had the
time to let this fragility slip from our minds. We could do no better
than to cite here Dedekind's 'proof' of what is put forward blithely
as the 'theorem' of paragraph 66:
66. Theorem: There exist infinite systems.
Proof: My own realm of thoughts, i.e. the totality S of all things, which
can be objects of my thought, is infinite. For, if 5 signifies an element
of S, then is the thought 5', that 5 can be an object of my thought, itself
an element of S. If we regard this as transform ((s) of the element s,
then has the transformation (of S, thus determined, the property that
the transform S' is a part of S; and S' is certainly a proper part of S,
because there are elements in S (e.g. my own ego) which are different
from every such thought s' and therefore are not contained in S'.
Finally it is clear that, if Sl and S2 are different elements of S, their
transforms 51' and sz' are also different, that therefore the transforma-
tion (is a distinct (similar) transformation. Hence S is infinite, which
was to be proved.
4.17. Once our stupor dissipates (but it is of the same order as
that which grips us in reading the first propositions of Spinoza's
Ethics), we must proceed to a close examination of this proof of
4.18. Some technical specifics: The force of the proof lies in the
consideration of the correspondence between an 'object of my thought'
and the thought 'this is an object of my thought' - that is to say, the
correspondence between a thought and the thought of that thought,
or reflection - as a function operating between elements of the set of
my possible thoughts (in fact, we may as well identify a 'possible
object of my thought' with one possible thought). This function is
'distinct' (we would now say injective), because it possesses the prop-
erty (which biunivocal correspondences also possess) that two dis-
tinct elements always correspond via the function to two distinct
elements. Given two thoughts whose objects distinguish them from
each other, the two thoughts of these thoughts are distinct (they also
have distinct objects, since they think of distinct thoughts). Conse-
quently there is a biunivocal correspondence between thoughts in
general and thoughts of the type 'thought of a thought'. Or, if you
prefer, there is such a correspondence between thoughts whose object
is anything whatsoever and thoughts whose object is a thought. Now
this second set forms a proper part of the set of all possible thoughts,
since there are thoughts which are not thoughts of thoughts: the
striking example Dedekind gives is what he calls 'the ego'. Thus the
set of all my possible thoughts, being in biunivocal correspondence
with one of its proper parts, is infinite.
4.19. Dedekind's approach is a singular combination of Descartes'
Cogito and the idea of the idea in Spinoza.
The starting point is the very space of the Cogito, as 'closed' con-
figuration of all possible thoughts, existential point of pure thought.
It is claimed (but only the Cog ito assures us of this) that something
like the set of all my possible thoughts exists.
From Spinoza's causal 'serialism' (regardless of whether or not he
figured in Dedekind's historical sources) are taken both the existence
of a 'parallelism' which allows us to identify simple ideas by way of
their object (Spinoza says: through the body of which the idea is an
idea), and the existence of a reflexive redoubling, which secures the
existence of 'complex' ideas, whose object is no longer a body, but
another idea. For Spinoza, as for Dedekind, this process of reflexive
redoubling must go to infinity. An idea of an idea (or the thought of
a thought of an object) is an idea. So there exists an idea that is the
idea of the idea of the idea of a body, and so on.
All of these themes have to be in place in order for Dedekind to
be able to conclude the existence of an infinite system. There must
be a circumscribed 'place', representable under the sign of the One,
of the set of my possible thoughts. We recognise here the soul, the
'thinking thing' as paradigmatically established by Descartes, in its
existence and essence (pure thought), in the Cogito. An idea must be
identifiable through its object, so that two different ideas correspond
to two different objects: this alone authorises the biunivocal character
of the correspondence. And, ultimately, it must be that the reflexive
process goes to infinity, since, if it did not, there would exist thoughts
with no correspondent through the function, thoughts for which
there were no thoughts of those thoughts. This would ruin the argu-
ment, since it would no longer be established that to every element
of the set of my possible thoughts S there corresponds an element of
the set of my reflexive thoughts S'. Ultimately - above all, I would
say - there must be at least one thought that is not reflexive, that is
not a thought of a thought. This alone guarantees that S', the set of
reflexive thoughts, is a proper part of S, the set of my possible
thoughts. This time, we recognise in this fixed point of difference the
Cogito as such - what Dedekind calls 'my own ego'. That which does
not allow itself to be thought as thought of a thought is the act of
thinking itself, the '[ think'. The 'I think' is non-decomposable; it is
impossible to grasp it as a thought of another thought, since every
other thought presupposes it.
It is therefore no exaggeration to say that for Dedekind, ultimately,
number exists in so far as there is the Cogito as pure point of exis-
tence, underlying all reflection (specifically, there is an 'I think that 1
think'), but itself situated outside of all reflection. The existential
foundation of the infinite, and therefore of number, is what Sartre
calls the 'pre-reflexive Cogito'.
And here we discover a variant of Jacques-Alain Miller's thesis:
what subtends number is the subject. The difference is that, whereas
for Miller it is the 'process of engendering' of number that requires
the function of the subject, for Dedekind it is the existence of the
infinite as its place. The Fregean programme of the conceptual deduc-
tion of zero and the Dedekindian programme of the structural deduc-
tion of the infinite lead back to the same point: the subject, whether
as insistence of lack or as pure point of existence. To the Lacanian
subject can be ascribed the genesis of zero, to the Cartesian subject:
the existence of the infinite. As if two of the three great modern chal-
lenges of thinking number (zero, the infinite, the downfall of the
One), once the third is assumed in the guise of a theory of sets,
can only be resolved through a radical employment of that great
philosophical category of modernity: the subject.
4.20. I could simply say that, just as I am not enough of a Leibnizian
to follow Frege, I am equally neither Cartesian nor Spinozist enough
to follow Dedekind.
4.21. Against Dedekind's Spinozism: Far from the idea of an infinite
recurrence of the thought of a thought of a thought of a thought of
a thought, and so on, being able to found the existence of the place
of number, it presupposes it. In fact, we have no experience of this
type. Only the existence - and consequently the thought - of the
sequence of numbers allows us to represent, and to make a numerical
fiction of, a reflection which reflects itself endlessly. The very possibil-
ity of stating a 'thought' at, say, the fourth or fifth level of reflection
obviously relies on an abstract knowledge of numbers as a condition.
As to the idea of a reflection that 'goes to infinity', this obviously
contains precisely what we are trying to demonstrate, namely the
effect of infinity in thought: an effect whose only known medium is
the mathematics of number.
4.22. As regards questions of existence, Spinoza himself made certain
not to proceed as Dedekind does. He never sought to infer the exis-
tence of the infinite from the recurrence of ideas. It is, rather, precisely
because he postulated an infinite substance that he was able to estab-
lish that the sequence that goes from the idea of a body to ideas of
ideas of ideas, and so on, is infinite. For Spinoza, and he is quite justi-
fied in this, the existence of the infinite is an axiom. His problem is
rather 'on the other side', the side of the body (or, in Dedekind's
terms, that of the object). For, if there is a rigorous parallelism
between the chain of ideas and the chain of bodies, then there must
be, corresponding to the idea of an idea, the 'body of a body', and
we are unable to grasp what the reality of such a thing might be.
Dedekind evades this problem because the place of thinking he
postulates assumes Cartesian closure: the corporeal exterior, the
extensive attribute, does not intervene in it. But, in seeking to draw
from Spinozist recurrence a conclusive (and non-axiomatic) thesis on
the infinite, he produces only a vicious circle.
4.23. Against Dedekind's Cartesianism: It is essential to the proof
that every thought can be the object of a thought. This theme is
incontestably Cartesian: the 'I think' subtends the being of ideas in
general as a 'material' of thought, and it is clear that there is no idea
that cannot be a thinkable idea, that is to say (since we are speaking
of the set of my possible thoughts) virtually actualisable as object of
my thought. But obviously this excludes the possibility that 'it'I3
could think without my thinking that I think that thought, and
without it being even possible to do so. Dedekind is Cartesian in his
exclusion of the unconscious, which, since Freud, we know to think,
and to think in such a way that some of its thoughts can be defined
precisely as those that 1 cannot think. 'Unconscious thoughts' are
precisely those unable, at least directly, to become objects of my
More generally, it is doubtful, for a contemporary philosopher,
whether true thoughts, those that are included in a generic procedure
of truth, could ever be exposed as such in the figure of their reflection.
This would be to imagine that their translation onto the figure of
knowledge (which is the figure of reflection) is coextensive with them.
Now the most solid idea of contemporary philosophy is precisely not
to understand the process of truth except as a gap in knowledge. If
'thought' means: instance of the subject in a truth-procedure, then
there is no thought of this thought, because it contains no knowledge.
Dedekind's approach founders on the unconscious, and does not hold
firmly enough to the distinction between knowledge and truth.
4.24. Descartes himself is more prudent than Dedekind. He makes
certain not to infer the infinite from reflection, or from the Cogito as
such. He does not consider, in proving the existence of God, the
totality of my possible thoughts, as Dedekind does. On the contrary,
he singularises an idea, the idea of God; his local argument might be
contrasted with Dedekind's global, or set-theoretical, argument. Des-
cartes' problem is elsewhere, it is a Fregean problem: how do we pass
from concept to existence? For this, an argument positing a dispro-
portion between the idea and its place is necessary: the idea of the
infinite is without common measure with its place, which is my soul
- or, in Dedekind's terms, the set of my possible thoughts; because
this place, grasped in its substantial being, is finite. The singular idea
of the infinite must therefore 'come from elsewhere'; it must come
from a real infinity.
We can see how, in the end, Descartes' and Dedekind's positions
are reversed. For Dedekind, it is the place that is infinite, because it
must support reflection (the capacity of the Cogito) in its going to
infinity. For Descartes, it is the exterior of the place (God) that is
infinite, since the place of my thought, guaranteed in its being by the
Cogito, is finite, and is therefore not capable of supporting alone the
idea of the infinite. But, in seeking to break with the finitude of the
place, Dedekind forgets that this place could well be nothing but a
scene fabricated by an Other place, or that thought could well find
its principle only in a presupposition of infinite number, of which it
would be the finite and irreflexive moment.
4.25. Immanent critique: Dedekind's starting point is 'the realm of
all possible objects of my thought', which he immediately decides to
call system S. But can this domain be considered as a system, that is
to say, a set? Do the 'possible objects of my thought' form a set, a
consistent multiplicity, which can be counted as one (leaving aside
the thorny question of knowing what carries out this accounting of
my thoughts)? Isn't it rather an inconsistent multiplicity, in so far as
its total recollection is, for thought itself, precisely impossible? If one
admits the Lacanian identification of the impossible and the real,
wouldn't the 'system' of all possible objects of my thoughts be the
real of thought, in the guise of the impossibility of its counting-for-
one? Before establishing that the 'realm of all possible objects of my
thought' is an infinite system, then, we must establish that it is a
system (a set) at all.
4.26. In the same way in which Russell's paradox comes to spoil
Frege's derivation of number on the basis of the concept, the 'paradox'
of the set of all sets - a descendant of the former - comes to break
Dedekind's deduction of the existence of the infinite, and conse-
quently the deduction of the existence of N, the 'simply infinite' set
which is the place of number. Conceptually set out by Dedekind with
impeccable inferences, the place of number does not stand the test of
consistency, which is also that of existence.
4.27. Reasoning 'a la Dedekind': Any system whatsoever (a set),
grasped in abstraction from the singularity of its objects or, as Dede-
kind says, thought uniquely according to 'that which distinguishes'
these objects (thus, their simple belonging to a system and its laws),
is obviously a possible object of my thought. Consequently, within
the supposed system S of all possible objects of my thought must
figure, as a subsystem (subset), the system of all systems, the set of
all sets. By virtue of this fact, this system of all systems is itself a
possible object of my thought. Or, in simplified terms, the system of
all systems is a thought.
Now, this is an impossible situation. In fact, a fundamental prin-
ciple of Dedekind's demonstration has it that every thought gives rise
to a thought of this thought, which is different from the original
thought. So if there exists a thought of the set of all sets, there must
exist a thought of this thought, which is in S, the set of all my possible
thoughts. S is then larger than the set of all sets, since it contains at
least one element (the thought of the set of all sets) that does not
figure in the set of all sets. Which cannot be, since S is a set, and
therefore must figure as an element in the set of all sets.
Or, once again: considered as a set or system, S, the domain of all
the possible objects of my thought, is an element of the set of all sets.
Considered in its serial or reflexive capacity, S overflows the set of
all sets, since it contains the thought of that thought which is the set
of all sets. S is thus at once inside (or 'smaller than') and outside (or
'larger than') one of its elements: the thought of the set of all sets.
We must conclude then, excluding logical inconsistency, either that
the set of all sets, the system of all systems, is not a possible ob;ect
of my thought, even though we have just thought it; or, more reason-
ably, that the domain of all possible objects of my thought is not a
system, or a set. But, in that case, it cannot be used to support the
proof of the existence of an infinite system.
4.28. Reasoning more mathematically now: Suppose that the set of
all sets exists (which implies necessarily the existence as set of the
domain of all possible objects of my thought). Then, since it is a set,
we can separate (Zermelo's axiom, 2.12), as an existent set, all of the
elements that have a certain property in common. Take the property
'not being an element of itself'. By means of separation this time, and
therefore with the guarantee of existence already in place, we 'cut
out' from the set of all sets, which we suppose to exist, the set of all
the sets which do not belong to themselves. This set then exists, which
Russell's paradox tells us is impossible (admitting the existence of the
set of all sets which do not belong to themselves leads directly to a
formal contradiction, d. 2.11). So it is impossible that the set of all
sets should exist, and a fortiori that the domain of all my possible
thoughts could be a set.
4.29. Dedekind's attempt ultimately fails at the same point as did
Frege's: in the transition from concept to assertion of existence. And
at the root of the affair is the same thing: Frege and Dedekind both
seek to deduce from 'pure logic', or thought as such, not just the
operational rules of number, but the fact of its existence for thought.
Now, just like the empty set, or zero, the infinite will not be deduced:
we have to decide its existence axiomatically, which comes down to
admitting that one takes this existence, not for a construction of
thought, but for a fact of Being.
The site of number, whether we approach it, like Frege, 'from
below', on the side of pure lack, or, like Dedekind, 'from above',
from the side of infinity, cannot be established logically, by the pres-
sure of thought alone upon itself. There has to be a pure and simple
acknowledgement of its existence: the Axiom of the Empty Set founds
zero, and, as a result of this, the finite cardinals exist. The Axiom of
Infinity founds the existence of the infinite ordinals, and from there
we can return to the existence of finite ordinals. The challenges posed
to the moderns by the thinking of number cannot be met through a
deduction, but only through a decision. And what subtends this deci-
sion, as to its veridicality, relates neither to intuition nor to proof. It
relates to the decision's conformity to that which being qua being
prescribes to us. From the fact that the One is not, it follows, with
regard to zero and the infinite, that nothing can be said other than:
they are.
4.30. Nevertheless, we must give Dedekind immense credit for three
crucial ideas.
The first is that the best approach to number is a general theory
of the pure multiple, and therefore a theory of sets. This approach,
an ontological one, entirely distinguishes him from the conceptual or
logicist approach, as found in Frege.
The second is that, within this framework, we must proceed in
'ordinal' fashion, erecting a sort of universal series where number
will come to be grasped. Certainly, the theory of ordinals must be
removed from its overdependence on the idea of order, still very much
present in Dedekind. Because, as I objected to Jacques-Alain Miller,
there is no reason to presume that the being of number will be await-
ing us along the ordered route that we propose to it. The concept of
the ordinal must be still further ontologised, rendered less opera-
tional, less purely serial.
The third of Dedekind's great inspired ideas is that, to construct
a modern thinking of number, a non-Greek thinking, we must begin
with the infinite. The fact that it is vain to try to give this beginning
the form of a proof of existence is ultimately a secondary matter,
compared to the idea of the beginning itself. It is truly paradigmatic
to have understood that, in order to think finite number, natural
whole number, it is necessary first to think, and to bring into exis-
tence - by way of a decision that respects the historial nature of being,
in so far as our epoch is that of the secularisation of the infinite (of
which its numericisation is the first instance) - infinite number.
On these three points, Dedekind is truly the closest companion,
and in certain respects the ancestor, of the father - still misunderstood
- of the great laws of our thought: Cantor.
5.1. Peano's work is certainly not comparable in profundity or in
novelty either to Frege's or to Dedekind's. His success lies more in
the clarification of a symbolism, in the firm assurance of the connec-
tion between logic and mathematics, and in a real talent for discern-
ing and denoting the pertinent axioms. One cannot speak of number
without tackling the famous 'Peano axioms' at their source; they have
become the reference text for any kind of formal introduction on the
natural whole numbers.
5.2. Even though, from the very beginning of his Principles of Arith-
metic,1 - written, deliciously, in Latin - Peano speaks of 'questions
that pertain to the foundations of mathematics', which he says have
not received a 'satisfactory solution',2 the approach he adopts is not
so much a fundamental meditation as a 'technicisation' of proce-
dures, with a view to establishing a sort of consensus on manipulation
(something in which, in fact, he succeeds perfectly). This is the sense
in which we ought to understand the phrase: 'The difficulty has its
main source in the ambiguity of language.'3 To expound number in
the clarity of a language - an artificial clarity, certainly, but legible
and indubitable - this is what is at stake in Peano's work.
5.3. In terms of its content, the approach is modelled on Dedekind's.
We 'start' from an initial term, which, as with Dedekind, is not
zero but one. We 'put to work' the successor function (denoted in
Peano according to the additive intuition: the successor of n is written
n + 1). We rely heavily on induction, or reasoning by recurrence. But,
whereas Dedekind, who works in a set-theoretical framework, deduces
the validity of this procedure, in Peano it is treated purely and simply
as an axiom. We decide that:
- if 1 possesses a property,
- and if it is true that, when n possesses a property, then n + 1 also
possesses it,
- then, all numbers n possess the property.
Armed with this inductive principle and with purely logical axioms
whose presentation he has clarified, Peano can define all the classical
structures of the domain of whole numbers: total order and algebraic
operations (addition, multiplication).
5.4. The axiom of induction, or of recurrence, marks the difference
in thinking between Peano and Dedekind on the crucial issue of the
infinite. Treated as a simple operational principle, recurrence actually
permits legislation over an infinite totality without making mention
of its infinity.
It is clear that there is an infinity of whole numbers. To speak of
'all' these numbers therefore means to speak of an actual infinity. But
in Peano's axiomatic apparatus, this infinity is not introduced as such.
The axiom of recurrence permits us, from a verification (1 possesses
the property) and an implicative proof (if n possesses the property,
then n + 1 also possesses it), to conclude that 'all numbers possess
the property', without having to inquire as to the extension of this
'all'. The universal quantifier here masks the thought of an actual
infinity: the infinite remains a latent form, inscribed in the quantifier
without being released into thought.
Thus Peano introduces the concept of number without transgress-
ing the old prohibition on actual infinity, a prohibition that still hangs
over our thought even as the latter is summoned to its abolition by
the modern injunction of being. Peano's axiomatic evades the infinite,
or explicit mention of the infinite.
For Dedekind, on the other hand, not only the concept of the
infinite, but also its existence, is absolutely crucial. Dedekind says
this explicitly in a letter to Keferstein:
After the essential nature of the simply infinite system, whose abstract
type is the number sequence N, had been recognized in my analysis
... the question arose: does such a system exist at all in the realm of
our ideas? Without a logical proof of existence it would always remain
doubtful whether the notion of such a system might not perhaps
contain internal contradictions. Hence the need for such proofs.
5.5. Peano does not broach questions of existence. When a system
of axioms is applied to operational arrangements, we will be able, if
necessary, to enquire as to that system's coherence; we need not
speculate on the being of that which is interrogated. The vocabulary
of the 'thing', or object, common to Frege and Dedekind (even if it
is a matter of 'mental things' in the sense of Husserl's noematic cor-
relate) is dropped in Peano's work, in favour of a somewhat 'post-
modern' sensibility where the sign reigns. For example, he writes: 'I
have denoted by signs all ideas that occur in the principles of arith-
metic, so that every proposition is stated only by means of these
signs.,5 If the latent model in Dedekind and of Frege is philosophical
('philosophy as rigorous science'),6 in Peano it is directly algebraic:
'With these notations, every proposition assumes the form and the
precision that equations have in algebra ... the procedures are similar
to those used in solving equations.'7
The 'economy of number' proposed by Peano is an economy of
signs whose paradigm is algebraic, whose transparency is consensual,
and whose operational effectiveness is therefore not in doubt. He thus
participates forcefully in that movement of thought, victorious today,
that wrests mathematics from its antique philosophical pedestal and
represents it to us as a grammar of signs where all that matters is the
making explicit of the code. Peano prepares the way from afar - by
eliminating all idea of a being of number, and, even more so, that of
number as being - for Carnap's major theses, which reduce mathe-
matics, treated as a 'formal language' (as opposed to empirical lan-
guages), not to a science (because according to this conception every
science must have an 'object'), but to the syntax of the sciences. Peano
is inscribed in the twentieth century's general movement of thought
- forged, in fact, at the end of the nineteenth century - whose char-
acteristic gesture is the destitution of Platonism, in the guise of that
which had always been its bastion: mathematics, and especially the
Idea of number.
5.6. We see here, as if in the pangs of its birth, the real origin of
what Lyotard calls the 'linguistic turn' in Western philosophy, and
what I call the reign of the great modern sophistry: if it is true that
mathematics, the highest expression of pure thought, in the final
analysis consists of nothing but syntactical apparatuses, grammars of
signs, then a fortiori all thought falls under the constitutive rule of
It is certain that, for Plato, the subordination of language to 'things
themselves', as dealt with for example in the Cratylus, has as its
horizon of certitude the ontological vocation of the matheme. There
is no upholding the pure empire of the sign if number, which we
indicate with just a simple stroke, is, as Plato thought, a form of
Being. Conversely, if number is nothing but a grammar of special
signs, ruled by axioms with no foundation in thought, then it becomes
probable that philosophy must be, first and foremost (as in Deleuze's
reading of Nietzsche's 'diagnostics'), a thinking of the force of signs.
Either truth or the arbitrariness of the sign and the diversity of syn-
tactical games: this is the central choice for contemporary philosophy.
Number occupies a strategic position in this conflict, because it is
simultaneously the most generalised basis of thought and that which
demands most abruptly the question of its being.
Peano's axiomatic, poor in thought but strong in its effects, a
grammar which subdues number, the organising principle of an oper-
ational consensus, a deft mediation of the infinite into the finitude of
signs, represents something of a lucky find, a gift, for modern
5.7. Every purely axiomatic procedure introduces undefined signs,
which can only be presented by codifing their usage in axioms. Peano
is hardly economical with these 'primitive' signs: there are four, in
fact (you are reminded that set theory has recourse to one single
primitive sign E, belonging, which denotes presentation as such):
Among the signs of arithmetic, those that can be expressed by other
signs of arithmetic together with the signs of logic represent the ideas
that we can define. Thus, I have defined all signs except four ... If, as
I think, these cannot be reduced any further, it is not possible to define
the ideas expressed by them through ideas assumed to be known
These four irreducible signs are:
1 The sign N, which 'means number (positive whole number}'.
2 The sign 1, which 'means unity'.
3 The sign a + 1, which 'means the successor of a'.
4 The sign =, which 'means is equal to'.
Peano thus explicitly renounces all definition of number, of suc-
cession, and of 1. (The case of the sign = might be treated separately:
it is in point of fact a logical sign, not an arithmetical one. Peano
himself writes: 'We consider this sign as new, although it has the form
of a sign of 10gic.')JO Evidently this is the price to be paid for opera-
tional transparency. Where Frege musters all thought towards
attempting to understand the revolutionary statement 'zero is a
number', Peano simply notes (it is the first axiom of his system): 1 E
N, a formal correlation between two undefined signs that 'means'
(but according to what doctrine of meaning?) that 1 is a number.
Where Dedekind generates the place of number as the space of pos-
sible employment, or the really existing infinite chain, of a biunivocal
function, Peano notes:
a E N a + 1 E N, an implication that
involves three undefined signs, and which 'means' that, if a is a
number, its successor is also a number. The force of the letter is here
at the mercy of meaning. And the effect is not one of obscurity,
but rather one of an excessive limpidity, a cumbersome levity of
the trace.
5.8. In the poem, the obscure is born of that which, as a breaking
open of the signifier at the limits of language, disseminates the letter.
In Peano's pure axiomatic, the retreat of sense issues from the fact
that the force of the letter is turned back upon itself, and that it is
only from outside that thought can come to it. Peano wishes to put
off any confrontation with the latent poem the absence of which
number - astral figure of being ('cold with neglect and disuse, a
Constellation') 12 - unfailingly instigates and the effect of which Frege
and Dedekind unconsciously preserve in the desperate attempt to
conjure forth into Presence now zero, now the infinite.
5.9. Peano's axiomatic is a shining success story of the tendency of
our times to see nothing in number except for a network of opera-
tions, a manipulable logic of the sign. Number, Peano thinks, makes
signs about the sign, or is the Sign of signs.
From this point of view, Peano is as one with the idea that the
universe of science reaches its apex in the forgetting of being, homog-
enous with the reabsorption of numericality into the unthought of
technical will. Number is truly machinic. Thus it can be maintained
that the success of Peano's axiomatic participates in the great move-
ment that has given up the matheme to modern sophistry, by unbind-
ing it from all ontology and by situating it within the sole resources
of language.
5.10. It will be a great revenge upon this operation to discover, with
Skolem and then Robinson,13 the semantic limits of the grammar of
signs to which Peano had reduced the concept of number. We know
today that such an axiomatic admits of 'non-standard' models, whose
proper being differs greatly from all that we intuitively understand
by the idea of natural whole number. So that Peano's system admits
of models where there exist 'infinitely large' numbers, or models
whose type of infinity exceeds the denumerable. Peano arithmetic is
susceptible to 'pathological' interpretations; it does not have the
power to establish a univocal thought within the machinism of signs.
Every attempt to reduce the matheme to the sole spatia Ii sed evidence
of a syntax of signs runs aground on the obscure prodigality of being
in the forms of the multiple.
5.11. The essence of number will not be spoken, either as simple
force of counting and of its rules, or as sovereignty of graphisms. We
must pass into it through a meditation on its being.
N is not an 'undefined' predicate, but the infinite place of exercise
of that which succeeds the void (or zero), the existential seal which
strikes there where it
insists on succeeding.
What 'begins' is not the 1 as opaque sign of 'unity', but zero as
suture of all language to the being of the situation whose language
it is.
Succession is not the additive coding of a + 1, but a singular dis-
position of certain numbers which are successors rather than their
succeeding, and which are marked in their being by this disposition.
We must know also that zero and the infinite are precisely that which
does not succeed, and that they are so in their very being, in different
ways; although both are located, by virtue of this fact, on the shores
of a Nothingness.
Number is neither that which counts, nor that with which we
count. This regime of numericality organises the forgetting of number.
To think number requires an overturning: it is because it is an unfath-
omable form of being that number prescribes to us that feeble form
of its approximation that is counting. Peano presents the inscription
of number, which is our infirmity, our finitude, as the condition of
its being. But there are more things, infinitely more, in the kingdom
of Number, than are dreamt of in Peano's arithmetic.
Cantor: 'Well-Orderedness'
and the Ordinals
6.1. The ordinals represent the general ontological horizon of numer-
icaliry. Following the elucidation of the concept of the ordinal, with
which we shall presently occupy ourselves, this principle will govern
everything that follows, and it is well said that in this sense Cantor
is the true founder of the contemporary thinking of number. In fact,
considered that the theory of ordinals constituted the very
heart of his discovery. Today, the working mathematician, for whom
it suffices that there are sets and numbers and who does not worry
at all about what they are, thinks of the ordinals rather as something
of a curiosity. We must see in this mild disdain one of the forms of
submission of the mathematician, in so far as he or she is exclusively
working, to the imperatives of social numericality. Specialists in
mathematical logic or set theory are doubtless an exception, even if
they themselves often regret this exception: in spite of themselves,
they are closest to the injunction of Being, and for them the ordinals
are essential.
6.2. I have said, in connection with Dedekind, that, in our present
philosophical discourse, we must assume as complete an 'ontologisa-
tion' of the ordinals as possible. The presentation of this concept
by Dedekind or Cantor relates it essentially to the notion of weIl-
orderedness - something still very close to a simple serial or
operational intuition of number.
6.3. Every schoolboy knows that, given two different whole numbers,
one of them is larger and the other smaller. And he knows also that,
given a 'bunch' of numbers, there is one and one only that is the
smallest of the bunch.
From this serial knowledge, if one abstracts out its general proper-
ties, the concept of the well-ordered set can be developed.
6.4. A 'well-ordered' set is a set for which:
• between the elements of the set, there is a relation of total order;
given two elements, e and e', if < denotes the order-relation, then
either e < e', e' < e, or e = e'; no two elements are 'non-comparable'
by this relation;
• given any non-empty part of the set so ordered, there is a smallest
element of this part (an element of this part that is smaller than
all the others). If P is the part considered, there exists p, which
belongs to P and for which, for every other p' belonging to P, P
< p' This element p will be called the minimal element of P.
If an element p is minimal for a part P, it alone possesses that
property. For, if there were another, a p' different from p, then, because
the order is total, either p < p' and p' would not be minimal, or p' <
p and p would not be minimal. So we can speak without hesitation
of the 'minimal element' of a part P of a well-ordered set.
We can see that the general concept of the well-ordered set is
merely a sort of extrapolation from what the schoolboy observes in
the most familiar numbers: the natural whole numbers.
6.5. A good image of a well-ordered set is as follows. Let E be such
a set. 'Start' with the smallest element of E, which, given condition
2 above, must exist. Call this element 1. Consider the part of E
obtained by removing 1, the part (E - 1). It too has a minimal
element, which comes in a certain sense straight after 1. Call this
element 2. Consider the part of E obtained by removing 1 and 2 to
be the part (E - (1,2)). It has a minimal element, call it 3, and so on.
A well-ordered set presents itself like a chain, so that every link of
the chain follows ('follows' meaning: comes just after in the relation
of total order) only one other, well determined (it is the minimal
element of what remains).
6.6. Cantor's stroke of genius was to refuse to limit this image to
the finite, and thereby to introduce infinite numerations. He had the
following idea: If I suppose the existence - beyond that sequence
1,2,3, ... , n, n + 1, ... - of a whole number which is the 'first' well-
ordered set, the matrix of all others, an 'infinite ordinal number' (0,
and declare it larger than all the numbers that precede it, then what
prevents me from continuing? I can very well treat 0) as the minimal
element of a well-ordered set that comes in some sense after the set
of all the whole numbers. And I can then consider the 'numbers'
0) + 1, 0) + 2, ... , 0) + n, ... , etc. I will arrive eventually at 0) + 0),
and will continue once again. No stopping point is prescribed to me,
so that I have a sort of total series, each term of which is the possible
measure of every existent sequence. This term indicates to me that,
however many came before it, it numbers every series of the
same length.
6.7. Allow me to call ordinal the measure of the length of a well-
ordered set, from its minimal element to its 'end'. The 'entire' sequence
of ordinals would then provide us with a scale of measurement for
such lengths. Each ordinal would represent a possible structure of
well-orderedness, determined by the way in which the elements
succeed each other, and by the total number of these elements. This
is why we say that an ordinal, whether finite (the ordinals which
come before 0), and which are quite simply the natural whole numbers)
or infinite (those ordinals which come after 0)), numbers a 'type of
well-orderedness' .
6.8. To give a technical grounding for this idea, we will consider the
class of well-ordered sets that are isomorphic to one of the sets among
them (and therefore isomorphic to each other). What should we
understand by this?
Take two well-ordered sets, E and E', < the order-relation of E,
and <' the order-relation of E'. I will say that E and E' are isomorphic
if there exists a biunivocal correspondence ((d. 4.5) between E and
E', such that, when el < e2, in E, then ((ell <' ((e2) in E'.
We can see that (projects the order of E into the order of E', and,
what's more, since (is biunivocal, there are 'as many' elements in E'
as in E. We can therefore say that E and E', considered strictly from
the point of view of their well-orderedness, and abstracted from the
singularity of their elements, are identical: the 'morphism' (form) of
their well-orderedness is 'iso' (the same), as the correspondence (
assures us.
In fact, each class of well-ordered sets isomorphic to each other
represents a well-orderedness, that well-orderedness common to all
sets of that class. It is this well-orderedness that can be represented
by an ordinal.
Thus an ordinal is the mark of a possible figure (a form, a
morphism) of well-orderedness, isomorphic to all the sets that
take that form. An ordinal is the number or the cipher of a
6.9. This conception, already moving strongly in the direction of
determining a horizon of being for all number in the form of a uni-
versal scale of measurement for forms of well-orderedness, neverthe-
less presents some serious difficulties; the first among them technical,
the remainder philosophical.
6.10. The technical difficulties are three in number, three questions
which must be answered:
1 Which is the first term in the total series of ordinals, the initial
link that 'anchors' the whole chain? This is the conceptual ques-
tion of zero or the empty set, which alone is able to number
sequences of no length, sequences with no elements, the well-
orderedness that orders nothing. This is the question that caught
out Frege.
2 What exactly is the procedure of thought that allows us to suppose
a beyond of the sequence of finite whole numbers? What is the
gesture by which we pass beyond the finite, and declare co, the
first ordinal which will not be a natural whole number, the first
mark of a well-orderedness that describes the structure of a non-
finite set? This is the existential question of the infinite, upon
which Dedekind foundered.
3 Does the universal series of ordinals - the scale of measurement
of all length, whether finite or infinite, the totality of specifications
of well-orderedness - exist in the set-theoretical framework? Isn't
it - like the 'system of all the possible objects of my thought'
introduced by Dedekind - an inconsistent totality, one that
thought cannot take as one of its possible objects? This is the
question of counting for one an 'absolute' totality. It is thus
the problem of the defection of the One as soon as we claim to
'count' the universe of discourse.
And so, once again, we find ourselves returned to the three chal-
lenges of the modern thinking of number: zero, the infinite and the
non-being of the One.
6.11. It rapidly turns out that the third problem admits of no positive
solution. Something that was at one time put forward as a 'paradox',
the Burali-Forti paradox, can actually be proved: the ordinals do not
form a set, they cannot be collected in a multiple that can be counted
for one. The idea of 'all' the ordinals is inconsistent, impossible; it
is, to this extent, the real of the horizon of the being of number.
This proof is very closely related to that which refutes Dedekind's
attempt to prove the existence of an infinite set (compare 4.28): the
set of 'all' the ordinals must itself be an ordinal, and thus it would
be inside itself (since it is a set of all the ordinals) and outside itself
(since it is not counted in the sequence it totalises). We are therefore
prohibited to speak of a 'set of ordinals' with no further qualification.
Which is precisely to say: 'being an ordinal' is a property with no
extension. It is possible to confirm that a certain object is an ordinal
(possesses the property), but not to count for one all the objects that
have this property.
6.12. I have said enough, in my critique of Frege and Dedekind, for
the treatment of problems 1 and 2 (6.10) to be anticipated: the exis-
tence of zero, or the empty set, and the existence of an infinite set
can in no way be deduced from 'purely logical' presuppositions. They
are axiomatic decisions, taken under the constraints of the historial
injunction of being. The world of modern thought is nothing other
than the effect of this injunction. Beginning in the Renaissance, by
way of a rupture with the Greek cosmos,2 it became necessary,
in order to be able to think at all in accordance with our pre-
understanding of ontological exigency, to assume:
• that the proper mode under which every situation 'that is' is
sutured to its being is not Presence, the dehiscence of that which
pro-poses itself within its limits, but pure subtraction, the unquali-
fiable void. In that form of being which is number, this can be
stated as follows: 'zero exists', or, in a style more homogenous
with Cantor's ontological creation: 'a set exists which has no
• that, in their quasi-totality, and by way of a rupture with the
mediaeval tradition which reserves this attribute for God alone,
situation-beings are infinite; so that, far from being a predicate
whose force is that of the sacred, the infinite is a banal determina-
tion of being, such as it proffers itself as pure multiplicity under
the law of a count-for-one. In that form of being which is number,
this can be stated as follows: 'an infinite set exists'; or, more
technically: 'an ordinal exists which is not a natural whole
number'. Or, in other words, Oro exists'.
6.13. One had to wait practically until the beginning of the twentieth
century before these decisions relating to zero and to the infinite
would be recognised in themselves (under the names of the Axiom
of the Empty Set and the Axiom of Infinity), although they had been
operative in thought for three hundred years. But this is not surpris-
ing. We can observe a veritable philosophical desperation constantly
putting these imperatives into reverse, whether through the intellec-
tual dereliction of the theme of finitude or through nostalgia for the
Greek ground of Presence. It is true that, when we are dealing with
pure declarations, decided in themselves, these declarations exhibit
the fragility of their historicity. No argument can support them.
What's more, certain truth procedures, in particular politics, art and
love, are not yet capable of sustaining such axioms, and so in many
ways remain Greek. They cling to Presence (art and love), continually
recusing the statement 'zero is the proper numeric name of being' in
order to give tribute to the obsolete rights of the One. Or (politics)
they manage finitude, corroding day after day the statement 'the
situation is infinite', in order to valorise the corrupted authority
of practicalities.
6.14. The two axioms of the void and of the infinite structure the
entire thinking of number. The pure void is that which supports there
being number, and the infinite, that by which it is affirmed that
number is the measure of the thinking of every situation. The fact
that this is a matter of axioms and not of theorems means that the
existence of zero and of the infinite is prescribed to thought by being,
in order that thought might exist in the ontological epoch of such an
In this sense, the current strength of reactive, archaic and religious
wills is necessarily accompanied by an irremediable opacity of number
- which, not ceasing to rule over us, since this is the epochal law of
being, nevertheless becomes unthinkable for us. Number may exist
as form of being but, as a result of the total secularisation of the void
and of the infinite, thought can no longer exist in the form and with
the force that the epoch prescribes to it. So number will now manifest
itself, without limit, as tyranny.
6.15. The principal philosophical difficulty ofthe Cantorian concept
of the ordinals is as follows. In the presentations which bind it to
the concept of well-orderedness, the theory of ordinals rather
seems to 'generalise' the intuition of natural whole number that
allows us to think the being of number. It draws its authority
from that which it claims to elucidate. The idea of well-orderedness
in effect does not so much found the concept of number as deduce
it from the lacunary and finite experience of numerical immediacy,
which I incarnated (in 6.3) in the sympathetic figure of the
If we truly wish to establish the being of number as the form of
the pure multiple, to remove it from the schoolroom (which means
also to subtract the concept from its ambient numericality), we must
distance ourselves from operational and serial manipulations. These
manipulations, so tangible in Peano, project onto the screen of modern
infinity the quasi-sensible image of our domestic numbers, the 1, fol-
lowed by 2, which precedes 3, and then the rest. The establishing of
the correct distance between thought and countable manipulations is
precisely what I call the ontologisation of the concept of number.
From the point at which we presently find ourselves, it takes on
the form of a most precise task: the ontologisation of the 'universal'
series of the ordinals. To proceed, we must abandon the idea of
well-orderedness and think ordination, ordinality, in an intrinsic
It is not as a measure of order, nor of disorder, that the concept
of number presents itself to thought. We demand an immanent deter-
mination of its being. And so for us the question now formulates
itself as follows: which predicate of the pure multiple, that can be
grasped outside of all serial engenderment, founds numericality? We
do not want to count; we want to think the count.
Concepts: Natural Multiplicities
Transitive Multiplicities
7.1. What permits the abandonment of every primitive bond between
number and order or seriality is the concept of the transitive set.
Only this structural - and essentially ontological - operator enables
an intrinsic determination of number as a figure of natural being. In
virtue of it, we are no longer trapped in the quandaries of the deduc-
tion of the concept (Frege), of the subject as causality of lack in serial
engenderment (Miller), of the existence of the infinite (Dedekind), or
of the 'schoolboy' intuition of well-orderedness (Cantor).
7.2. Although this concept might seem at first glance rather mysteri-
ous, its lack of relation to any intuitive idea of number is to my eyes
a great virtue. It proves that in it we grasp something that breaks the
circle of an ontological elucidation of number entirely transparent in
its pure and simple presupposition. We have seen that this circle
recurs in Frege and in Dedekind, and that the Cantorian conception
of ordinals as types of well-orderedness is still compliant with it. But
we shall see that the legitimacy for philosophical thought of the
concept of transitivity leaves no room for doubt.
7.3. To understand what a transitive set is, it is essential to penetrate
the distinction - of which it would not be an exaggeration to say that
it supports all post-Cantorian mathematics - between an element's
belonging to a set and the inclusion of a part. This distinction is
rudimentary, but it implies such profound consequences that for a
long time it remained obscure.
7.4. A set is 'made out of elements', is the 'collection' (in my lan-
guage, the count-for-one) of its elements.
Take the set E, and let e be one of the elements from which it
'makes' a set: we denote this by e E E, and we say that e belongs to
E, E being the sign for belonging.
If you now 'gather together" many elements of E, they form a part
of E. Taking E' as the set of these elements, E' is a part of E. This is
denoted by E' c E, and we say that E' is included in E, c being the
sign of inclusion.
Every element of a part E' of E is an element of E. In fact this is
the definition of a part: E' is included in E when all the elements that
belong to E' also belong to E. So we see that inclusion is defined in
terms of belonging, which is the only 'primitive' sign of set theory.
The classic (misleading) image is drawn like this:
In it we can see that E' is a part of E, that e, is at once (as is every
element of E') an element of E' and an element of E, and that
e2 is an element of 'the whole' E, but not of the part E'. We also say
that e2 belongs to the difference of E and E', which is denoted by
7.5. Is it possible for an element that belongs to set E also to be a
part of that set, also to be included? This seems totally bizarre, above
all if we refer to the image above. But this sentiment misses the most
important point, which is that an element of a set can obviously be
itself a set (and even that this is always the case). Consequently, if e
belongs to E, and e is a set, the question arises whether an element
of e is or is not, in its turn, an element of E. If all the elements of e
are also elements of E, then e, which is an element of E, is also a part
of E. It belongs to E and is included in E.
7.6. Suppose for example that V is the set of living beings. My cat
belongs to this set. But a cat is composed of cells, which one might
say are themselves all living beings. So my cat is at once a living being
and a set of living beings. He belongs to V (qua one, this living cat),
and he is a part of V - he is included in V (qua group of living
7.7. Forget cats. Consider the three following 'objects':
- the object e];
- the object ez;
- the object which is the 'gathering together' of the first two, and
which we denote by (e],e2)' This is called the pair of e] and ez.
Form a set from these three objects. In the same way, we denote
it by: (e],e2,(e],e2))' This is called the triplet of e] and e2 and the pair
(e],ez). We will denote it by T. Note that the three elements that
belong to this triplet are e], e2, and (e],ez).
Since e] and ez belong to T, if I 'gather them together', I obtain a
part of T. Thus, the pair (e],ez), which is the 'gathering together' of
these two elements of T, is included in T. But in addition we can see
that it is an element of it, that it also belongs to it. Thus we have
constructed a very simple case of a set of which an element is also a
part. In set T, the pair (e],ez) is simultaneously in a position of belong-
ing and of inclusion.
7.8. We know, from a famous theorem of Cantor's, that there are
more parts than elements in any set E whatsoever. This is what I call
the excess of inclusion over belonging, a law of being qua being
whose consequences for thought are immense, since it affects the
fundamental categories that inform the couplets One/Multiple and
WholelPart. It is therefore impossible that every part should be an
element, that everything that is included should also belong: there are
always parts that are not elements.
But we can put the question from the other direction: since we can
see that it is possible in cenain cases (for example my cat for the set
V of living beings, or the pair (e],ez) for our triplet T) for an element
to be a part, is it possible for all elements to be parts, for everything
that belongs to the set to be included? This is not the case for T: the
element e] taken alone, for example, is not a part of T.
Can we produce a non-empirical example (because my V, my cat
and its cells are rationally suspect) of a set all of whose elements
would be parts?
7.9. Let's retrace our steps a little, back to the empty set. We have
proposed (in 2.18) the axiom 'a set exists which has no elements',
that is, a set to which nothing belongs. We are going to give to this
set, the 'empty' rock of the whole edifice of multiple-being, a proper
name, the name '0'.
The following, extremely subtle, remark must be made: the empty
set is a part of every set; 0 is included in E whatever E might be. Why?
Because, if a set F is not a part of E, it is because there are elements
of F that are not elements of E (if every element of F is an element of
E, then by definition F is a part of E). Now 0 has no elements. So, it
is impossible for it not to be a part of E. The empty set is 'universally'
included, because nothing in it can prevent or deny such inclusion.
To put it another way: to demonstrate that F is not a part of E
requires that we pick out, within F, at least one element: that element
which, not being an element of E, proves that F cannot be included
'entirely' within E. Now the void does not tolerate any differentiation
of this sort. It is in-different, and, because of this, it is included in
every multiplicity.
7.10. Consider the two following 'objects':
- the empty set, 0;
- the set whose one and only element is the empty set, which is
called the singleton of the empty set, and is denoted by (0).
Note well that this second object is different from the empty set
itself. In fact, the empty set has no elements, whereas the singleton
has one element - precisely, the empty set. The singleton of the
void 'counts for one' the void, whereas the empty set does not count
anything (this indicates a subtle distinction between 'does not
count anything', which is what 0 does, and 'counts nothing', which
is what (0) does. Plato already played on this distinction in the
7.11. An additional remark as regards singletons (singletons
'in general', not the particular singleton of the empty set): Take a set
E and one of its elements e (so e E E). The singleton of e, written (e),
is a part of E: (e) c E.
What is the singleton of e, in fact? It is the set whose unique
element is e. Consequently, if e is an element of E, 'all' the elements
of the singleton (e) - namely the single element e - are elements of
E, and so (e) is included in E.
7.12. 'Gather together' our two ohjects, the empty set denoted by 0
and the singleton of the empty set, denoted by (0). We obtain the
pair (0, (0)), which we will denote by D. This time, the two elements
of the pair D are also parts; everything that belongs to D is also
included in D. In fact, the first element, 0, the empty set, is included
in any set whatsoever (see 7.9). Specifically, it is a part of the pair
D. But, what's more, since 0 is an element of D, its singleton (0), is
a part of D (7.11). But (0) is precisely the second element of D. Thus
this element is also included in D. The set D is such that every element
of it is also a part; everything that belongs to D is included in D.
7.13. As predicted by Cantor's theorem, there are parts of D that are
not elements of D. For example, the singleton of the element (0) of
D is a part of D, as is every singleton of an element (7.11). We can
write this 'singleton of the singleton' as ((0)). Now, this object is not
one of the two elements of D.
7.14. An important definition: we say that a set T is transitive if it
is like the set D that we have just built: if all of its elements are also
parts, if everything that belongs to it is also included in it, if, wherever
it is the case that t E T, it is also the case that t c T.
7.15. Transitive sets exist, without a doubt. Perhaps V, the set of
living beings; certainly the set (0,(0)), which is transparent, translu-
cent even, constructed as it is from the void (the pair of the void and
the singleton of the void, the void as such and the void as one).
7.16. Modernity is defined by the fact that the One is not (Nietzsche
said that 'God is dead', but for him the One of Life took the place
of the deceased). So, for we moderns (or 'free spirits'), the Multiple-
without-One is the last word on being qua being. Now the thought
of the pure multiple, of the multiple considered in itself, without
consideration of what it is the multiple of (so: without consideration
of any object whatsoever), is called: 'mathematical set theory'. There-
fore every major concept of this theory can be understood as a
concept of modern ontology.
What does ontology discover in the concept of the transitive
7.17. Belonging is an ontological function of presentation, indicating
that which is presented in the count-for-one of a multiple. Inclusion
is the ontological function of representation, indicating multiples re-
counted as parts in the framework of a representation. A most impor-
tant problem (the problem of the state of a situation) is determined
by the relation between presentation and representation.
Now, a transitive set represents the maximum possible equilibrium
between belonging and inclusion, the element and the part, E and c.
Transitivity thus expresses the superior type of ontological stability;
the strongest correlation between presentation and representation.
There is always an excess of parts over elements (Cantor's theorem),
there always exist parts of a set which are not elements of that set.
Thus we obtain the maximal correspondence between belonging and
inclusion precisely when every element is a part: when the set con-
sidered is transitive.
This strong internal frame of the transitive set (the fact that every-
thing that it presents in the multiple that it is, it represents a second
time in the form of inclusion), this equilibrium, this maximal stability,
has led me to say that transitive sets are 'normal', taking 'normal' in
the double sense of non-pathological, stable, strongly equilibriated,
that is to say: not exposed to the disequilibrium between presentation
and representation, a disequilibrium whose effective form is the
evental caesura; and submitted to a norm, that of a maximally
extended correspondence between the two major categories of
ontological immanence: belonging and inclusion.
7.18. The concept of transitive multiplicity will constitute the normal
basis for the thinking of number. Transitivity is at once that which
makes of number a section
taken from the equilibrated fabric of
being and that which provides the norm for this section.
Von Neumann Ordinals
8.1. Let's consider more closely set D, introduced in 7.12, written as
(0,(0)), which is the pair of the void and the singleton of the void.
We know that set D is transitive: its two elements, 0 and (0),
are also parts of D. We can make a further remark here: these two
elements are also transitive sets.
• That (0) is transitive is self-evident: the only element of the single-
ton (0) is O. Now, 0 is a 'universal' part included in every set,
and, in particular, it is included in the set (0). So the unique
element of (0) is also a part of (0), and consequently (0) is a
transitive set.
• That 0, the empty set, is transitive results from its negative 'poros-
ity' to every property, which already makes it a part of any set
whatsoever (compare 7.9): a transitive set is one all of whose ele-
ments are also parts. Thus a set that is not transitive has at least
one element that is not a part. Now 0 has no elements. So it
cannot not be transitive. And, so, it is.
With our set D we have constructed not only a transitive set, but
a transitive set of transitive sets: this transitive set 'gathers together'
transitive sets. Both 0, (0) and their pair (0,(0)), are transitive.
8.2. A truly fundamental definition: A set is an ordinal (in von
Neumann's sense)' if it is like D, that is, if it is transitive and all of
its elements are transitive.
8.3. This definition completes the technical part of the ontologisation
of the concept of the ordinal. We are no longer dealing with well-
orderedness, with the image of the sequence of natural whole numbers,
or with an operational status. Our concept is purely immanent. It
describes a certain internal structural form of the ordinal, a form that
connects together in a singular fashion the two crucial ontological
operators belonging and inclusion, E and Co
Set D, which we have used as an exemplary case, is therefore an
ordinal. We can lift a corner of the veil on its identity: it is the number
Two. Moreover, this Two allows us to affirm that von Neumann
ordinals exist.
8.4. Before deploying this new concept of the ordinal, let's begin with
a first examination of the status of its definition and of the reasons
why the ordinals constitute the absolute ontological horizon of all
8.5. I have indicated (7.16) that a transitive set is the ontological
schema of the 'normal' multiple. Taking into account the fact that
the excess of representation over presentation is irremediable, tran-
sitivity represents the maximal equilibrium between the two.
Now, not only is an ordinal transitive, but all of its elements are
also transitive. An ordinal disseminates to the interior of a multiple
that normality which characterises it. It is a normality of normalities,
an equilibrium of equilibria.
A truly remarkable property results from this, which is that every
element of an ordinal is an ordinal.
Take an ordinaF W, and an element of that ordinal x (so that
x E W). W being an ordinal, all of its elements are transitive, so x is
transitive. For the same reason (the ordinality of W) W is itself transi-
tive, so x, an element of W, is also a part of W: x c W. As a result,
all the elements of x are elements of W. And, just as all the elements
of Ware transitive, the same follows for all the elements of x. The
set x is thus a transitive set all of whose elements are transitive: it is
an ordinal.
8.6. If transitivity is a property of stability, this time we discover a
complementary property of homogeneity: that which makes up the
internal multiple of an ordinal, the elements belonging to it, are all
ordinals. An ordinal is the count-for-one of a multiplicity of
Because of this homogenous and stable 'fabric' of ordinal multi-
plicity, I have been led to say that ordinals are the ontological schema
of the natural multiple. I call 'natural' (by way of opposition to
multiplicities that are unstable, heterogeneous, historical, and which
are thus exposed to the evental caesura) precisely that which is exem-
plified by the underlying multiple-being as thought by mathematics:
a maximal consistency, an immanent stability without lacuna, and a
perfect homogeneity, in so far as that of which this multiple-being is
composed is of the same type as itself.
We therefore posit, once and for all, that an ordinal is the index
of the being of a natural multiplicity.
8.7. If it is true that the ordinals constitute the great ontological
'ground' of number, then we can also say that number is a figure of
natural being, or that number proceeds from Nature. With the caveat,
however, that 'Nature' refers here to nothing sensible, to no experi-
ence: 'Nature' is an ontological category, a category of the thought
of the pure multiple, or set theory.
8.8. Must we say simultaneously that ordinals 'are numbers'? Such
would indeed be the idea of Cantor, who thought to achieve by way
of the ordinals an infinite prolongation of the sequence of whole
numbers. But for us, who have yet to propose any concept of number,
this would be begging the question. We will see, after having defined
what I call Number (the capitalisation is not for the sake of majesty,
but to designate a concept that subsumes all species of number,
known or unknown), that the ordinals, though playing a decisive role
in this definition, are only the representable amongst numbers, in the
numerical swarming which being lavishes on the ground of Nature.
The ordinals will thus be at once the instrument of our access to
number, of our thinking of number, and, albeit lost in a profusion
of Numbers that exceeds them in every way, they will be represent-
able or figurable as themselves, too, being Numbers.
8.9. The empty set, 0, is an ordinal. We have seen above that it is
transitive (8.1). It follows that all its elements are also transitive:
having no elements, how could it have an element that was not transi-
tive? Contrary to all intuition, zero, or the void, is a natural ontologi-
cal donation. The void, which sutures all language and all thought
to being, is also the point of nature where number is anchored.
8.10. Von Neumann ordinals have two crucial properties:
1 They are totally ordered by the fundamental ontological relation
belonging, the sign of multiple-presentation. That is to say that,
given two ordinals WI and W
, either the former belongs to the
latter (WI E W
), or the other way around (W2 E Wd, or they
are identical (WI = W2).
2 They obey a principle of minimality: given any property P what-
soever, if an ordinal possesses this property, then there exists a
smallest ordinal to possess it. Order is always belonging: if you
have an ordinal W such that it possesses the property P (if the
statement P(W) is true), then there exists an ordinal WI which
has the property and which is the smallest to have it (if W
E W b
doesn't have the property).
These two properties are natural. The first expresses the universal
intrication of those stable and homogenous multiplicities that are
natural multiplicities (see 8.6): thought in their being, two natural
multiples - two ordinals, then - cannot be independent. Either one
is in the presentation of the other, or vice versa. Nature does not
tolerate indifference or disconnection. The second property expresses
the 'atomic' or, if you like, 'quantum' character of nature. If a prop-
erty applies to some natural multiple, then there is always a natural
multiple that is the minimal support of that property.
Taken together, these two properties reunite the global status
of nature with its local status. Even though Nature
does not exist
(there is no set of all the ordinals, see 6.11), there is a sort of unity
of plan, of global interdependence, between natural multiples:
the presentation of which they are the schema is always 'embedded'.
And, although there are no unique and indiscernible components
of nature like the Ancients' atoms (unless one considers the void
as such), there is an exceptional local point for every property
that obtains for the 'regions' of nature: the minimal support of this
This articulation of the global and the local lends its ontological
framework to every Physics.
8.11. The two crucial properties (total order and minimality) can
both be proved on the basis of von Neumann's definition of the
These proofs depend upon a key principle of set theory (ontology
of the multiple): the Axiom of Foundation.
This axiom says that
every situation (every pure multiple) comprises at least one term (one
element) that has 'nothing in common' with the situation, in the
sense that nothing of that which composes the term (no element of
the element) is presented in the situation (belongs to the original
8.12. Let's return to the example of my cat (7.6). It is an element of
the set of living beings, and it is composed of cells that are in turn
elements of this set, if one grants that they are living organisms. But
if we decompose a cell into molecules, then into atoms, we eventually
reach purely physical elements that don't belong to the set of living
beings. There is a certain term (perhaps the cell, in fact) which
belongs to the set of living beings, but none of whose elements
belongs to the set of living beings, because those elements all involve
only 'inert' physico-chemical materiality. Of this term, which belongs
to the set but none of whose elements belongs to it, we can say
that it grounds the set, or that it is a fundamental term of the set.
'Fundamental' meaning that on one side of the term, we break through
that which it constitutes; we leave the original set, we exceed its
presentative capacity.
8.13. Once more, let's leave living beings, cats, cells and atoms
behind. Consider the singleton of the singleton of the void, that is,
the set whose unique element is the singleton of the void, and which
is written as ((0)). The element (0) of this set has as its only element
the void, O. Now the void is not an element of the original set ((0)),
whose only element is (0), because the void 0 and the singleton
of the void (0) are different sets. So (0) represents, in ((0)), a local
foundation-point: it has no element in common with the original set
((0)). That which it presents qua multiple -that is, 0 - is not presented
by ((0)), in the presentation in which it figures.
The Axiom of Foundation tells us that this situation is a law of
being: every multiple is founded, every multiple comprises at least
one element which presents nothing that the multiple itself
8.14. The Axiom of Foundation has a remarkable consequence,
which is that no set can belong to itself, that no multiple figures in
its own presentation, that no multiple counts itself as one. In this
sense, being knows nothing of reflection.
Take a set E which is an element of itself: E E E. Consider the
singleton of this set, (E). The only element of this singleton is E. So
E must found (E). But this is impossible, since E belongs to E, and
thus has in common with (E) that element which is itself. Since the
axiom of foundation is a law of being, we must reject the original
hypothesis: there does not exist any set that is an element of itself.
8.15. Returning to the crucial properties of the ordinals: They can
be proved, once the axiom of foundation is assumed. I will do so
here for the principle of minimality. For the principle of total order
through belonging, see the note.
Take an ordinal WI which possesses property P. If it is minimal,
all is well. Suppose that it is not. In that case, there exist ordinals
smaller than WI (and which therefore belong to WI, since the order
in question is belonging) and which also possess the property. Con-
sider the set E of these ordinals ('gathering together' all those which
possess property P and belong to Wd. Set E obeys the Axiom of
Foundation. So there is an element W
of E which is an ordinal (since
E is a set of ordinals) that possesses property P (since all the elements
of E possess it) and that has no element in common with E.
But, since W I is an ordinal, it is transitive. So W 2, which belongs
to it, is also a part of it: the elements of W
are all elements of WI.
If an element of W
possesses property P, then, since it is an element
of WI! it must belong to E (since E is the set of all the elements of
WI possessing property Pl. Which cannot be, because W
founds E
and therefore has no element in common with E. Consequently,
no element of W
has the property P, and W
is minimal for this
property. QED.
8.16. Thus is knitted the ontological fabric from which the numbers
will be cut out.
Homogenous, intricate, rooted in the void, locally
minimisable for every property, it is very much what we could call a
horizonal structure.
Succession and Limit.
The Infinite
9.1. In chapter 6, when we spoke of Dedekind's and Cantor's
approaches to the notion of the ordinal (on the basis of well-ordered-
ness), we saw that the whole problem was that after one ordinal
comes another, well-determined, and that this series can be pursued
indefinitely. We also saw that it was not at all the same thing to 'pass'
from n to n + 1 (its successor) as to pass from 'all' the natural
numbers to their beyond, which is the infinite ordinal (0. In the latter
case, there is manifestly a shift, the punctuation of a 'passage to the
In the ontologised concept of the ordinals which von Neumann
proposed and to which we dedicated chapter 8, do we once more
find this dialectic between simple succession and the 'leap' to the
infinite? And, more generally, how does the thorny issue of the exis-
tence of an infinite multiple present itself in this new context?
9.2. Let's apply ourselves firstly to the concept of succession.
We must take care here. The image of succession, of 'passage' to
the next, is so vividly present in the immediate representation of
number that it is often thought to be constitutive of its essence. I
reproached J. A. Miller (see 3.17) precisely for reducing the problem
of number to the determination of that which insists in its succes-
sional engenderment. I held that the law of the serial passage across
the numeric domain, a law which is imposed on us, does not coincide
with the ontological immanence of number as singular form of the
Consequently, if we find the idea of succession once again in von
Neumann's conception of the ordinals, it too must yield to the process
of ontologisation. Our goal will be to discover, not so much a prin-
ciple of passage as an intrinsic qualification of that which succeeds,
as opposed to that which does not. What counts for us is not succes-
sion, but the being of the successor. The repetitive monotony of
Peano's +1 does not concern us any longer: what we want to think
is the proper being of that which can only be attained in the modality
of the additional step.
9.3. Let's consider an ordinal W, in von Neumann's sense (a transi-
tive set all of whose elements are transitive).
A set, then, whose elements are:
- all of the elements of W;
- W itself.
So, to everything that composes the multiple W, we 'add' one
supplementary element, namely W itself. And it is indeed a question
of the adjunction of a new element, since we know (it is a conse-
quence of the axiom of foundation, compare 8.14) that W is never
an element of itself.
A non-operational form of +1 can be seen emerging here: it is not
a matter of an extrinsic addition, of an external 'plus', but of a sort
of immanent torsion, which 'completes' the interior multiple of W
with the count-for-one of that multiple, a count whose name is pre-
cisely W. The +1 consists here in extending the rule of the assembly
of sets to what had heretofore been the principle of this assembly,
that is, the unification of the set W, which is thereafter aligned with
its own elements, counting along with them.
9.4. An example of the procedure. We have demonstrated that set
D, which is written (0,(0)), and which is the pair of the void and the
singleton of the void, is an ordinal (it is transitive and all its elements
are transitive). Our non-operational definition of +1 consists in
forming the set of the three following elements: the two elements of
D and D itself. We write this as (0,(0),(0,(0))) (the 'whole' D is found
in the third position). Call this triplet T. We can now demonstrate
• T is transitive. Its first element, 0, is a universal part, and so it
must be a part of T; its second element, (0), is the singleton of its
first element, o. So it is also a part ofT (see 7.11). Its third element
(0,(0)) is nothing but the 'gathering together', the forming into a
pair, of these first two. So it is also a part. So every element of T
is a part, and T is transitive.
• All the elements of T are transitive. Given that we have shown
that D is an ordinal, we have duly shown that its elements, 0 and
(0), are also transitive. We have equally demonstrated that it itself,
(0,(0)), is transitive. And these are precisely the three elements
So T, obtained by 'adjoining' D to the elements of D, is a von
Neumann ordinal: a transitive set all of whose elements are
9.5. The reasoning we just followed can easily be generalised. For
any ordinal W whatsoever, everything will follow just as for T: the
set obtained in adjoining W itself, as an element, to W's elements is
an ordinal.
We 'step' from W to a new ordinal by adjoining to W's elements
a single additional element (this, now, allows us to lift a corner of
the veil on the identity of our example T: just as D was two - I would
like to say the being of number Two - T is none other than the
number Three).
The fact that one steps from W to a new ordinal, whose
elements are those of W supplemented by the one-name of their
assembly, by way of a sort of immanent +1, justifies the following
definition: we will call the ordinal obtained by joining W to the
elements of W, the successor of the ordinal W, and will denote it
by S(W).
So, in our example, T (three) is the successor of D (two).
9.6. The idea of the 'passage' from two to three, or from W to S(W),
is, in truth, purely metaphorical. In fact, from the start there are
figures of a multiple-being, D and T, and what we have defined is a
relation whose sole purpose is to facilitate for us the intelligible
passage through their existences. Finitude demands the binding of the
un-binding of being. We therefore think, in the succession T = S(D),
a relation whose basis is, in truth, immanent: T has the structural
property, verifiable in its ontological composition, of being the suc-
cessor of D, and it is no more than a necessary illusion to represent
T as being constructed or defined by the relation S, which connects
it externally to D.
A more rigorous philosophical approach consists in examining the
ordinals in themselves and in asking ourselves whether they possess
the property of succeeding. For example, T possesses the property of
succeeding D, recognisable in itself from the fact that D is an element
of T, and, what's more - as we shall see - that D is an element that
can be immanently distinguished (it is 'maximal' in T).
We will call successor ordinal an ordinal that possesses the prop-
erty of succeeding.
So T is a successor ordinal.
9.7. It might be objected that the property 'succeeds W' is still latent
in the intrinsic concept of successor, and therefore that we have failed
to establish ourselves in the ontological unbinding. This objection
can be alleviated.
Let's consider an ordinal W having the following, purely imma-
nent, property: amongst the elements of W, there is one element, say
wI. of which all the other elements of Ware elements: if Wz is an
element of W different from wI, then Wz E WI. I say that W is neces-
sarily a successor ordinal (in fact, it succeeds wd.
For if this situation obtains, it is because W's elements are:
- on the one hand the element WI;
- on the other, elements which, like wz, are elements of WI.
But, in reality, all the elements of WI are elements of W. For we
know that belonging, E, is a total order over the ordinals (see 8.10).
Now, all the elements of an ordinal are ordinals (8.5); specifically,
all the elements of Ware ordinals. WI is therefore an ordinal, and it
follows that the elements of WI are all ordinals. These elements are
connected to ordinal WI and W by the relation of total order that is
belonging: if WE WI, since WI E W, then w E W (transitivity of the
Thus W is composed of all the elements of WI, and WI itself: W is
by definition the successor of WI.
Let's agree to call the maximal element of an ordinal the
element of that ordinal which is like WI for W: all the other
elements of the ordinal belong to the maximal element. The
reasoning above now permits us to make the following definition:
An ordi1zal will be called a successor if it possesses a maximal
Here we are in possession of a totally intrinsic definition of the
successor ordinal. The singular existence of an 'internal' maximum,
located solely through the examination of the multiple structure of
the ordinal, of the fabric of elementary belonging at its heart, allows
us to decide whether it is a successor or not.
9.S. Since we now have an immanent, non-relational and non-serial
concept of 'what a successor is', we can pose the question: Are there
ordinals that are not successors?
9.9. The empty set, 0, is an ordinal that is not a successor. It obvi-
ously cannot succeed anything, since it has no elements and, to
succeed, it must have at least one element, namely the ordinal that it
Or, staying closer to the immanent characterisation: to be a suc-
cessor, ° must have a maximal element. Having no elements, it
cannot be a successor.
Once again, we discover the void's function as ontological anchor:
purely decided in its being, it is not inferable and, in particular, it
cannot succeed: the void is itself on the edge of the void, there is no
way it could follow from being, of which it is the original point.
9.10. All the ordinals that we have used in our examples, apart from
the void, are successors. Thus (0) (which is the number 1) is the suc-
cessor of 0. The number 2, whose being is (0,(0)), and which is
composed of the void and 1, is the successor of 1. And our T (the
number 3), which is composed of the void, 1, and 2 and is written
(0,(0),(0,(0))), is the successor of 2. It is clear that we can continue,
and will thereby obtain 4, 5, and, finally, any of the natural whole
numbers, all of which are successor ordinals.
9.11. Does this mean that we have at our disposal a thinking of
natural whole number? Not yet. We can say that 1, then 2, then 3,
etc., if we think each in its multiple-being, are natural whole numbers.
But, without being able to determine the place of their deployment,
it is impossible for us to pass beyond this case-by-case designation
and to propose a general concept of whole number. As Dedekind
perceived, such a concept necessitates a detour through the infinite,
since it is within the infinite that the finite insists. The only thing that
we can say with certainty is that whole numbers are successor ordi-
nals. But this is certainly not a sufficient characterisation of them:
there might well be other successors that are not whole numbers,
perhaps successors that are not even finite sets.
9.12. The question becomes: are there any other non-successor
ordinals apart from the void?
Let's call these non-successor ordinals different from ° (without
yet knowing whether they exist) limit ordinals. We ask once more:
do limit ordinals exist?
We are not yet in a position to decide upon this question. But we
can prove that, if they do, they are structurally very different from
successor ordinals.
9.13. No ordinal can come in between an ordinal Wand its successor
S(W). By this we mean that, given that the order-relation between
ordinals is that of belonging, no ordinal W I exists such that we have
the sequence W E WI E S(W).
We know in fact that W is the maximal element of S(W) (see 9.7).
Consequently, every element of S(W) that is different from W belongs
to W. Now, our supposed WI belongs to S(W). Therefore one of two
things must apply:
• either WI is identical to W. But this is impossible, because we
have supposed that W E WI. which would give us W E W. But
we know (8.14) that no set can be an element of itself;
• or WI is an element of W. But then it would not be possible that
WE WI. since WI E W.
It can be seen that ordinal succession is the schema of the 'one
more step', understood as that which hollows out a void between the
initial state and the final state. Between the ordinal Wand its succes-
sor S(W), there is nothing. Meaning: nothing natural, no ordinal. We
could also say that a successor ordinal delimits, just 'behind' itself, a
gap where nothing can be established. In this sense, rather than suc-
ceeding, a successor ordinal begins: it has no attachment, no continu-
ity, with that which precedes it. The successor ordinal opens up for
thought a beginning in being.
9.14. A limit ordinal, if such a thing exists, is a different case alto-
gether. The definition of such an ordinal is, please note, purely nega-
tive: it is not a successor; that is all that we know of it for the moment.
We can also say: it does not possess a maximal element. But the
consequences of this lack are considerable.
Take L, a supposed limit ordinal, and WI, an element of this
ordinal. Since WI is not maximal, there certainly exists an element Wl
of L which is larger than it: so we have the chain: WI E Wz E L. But,
since in its turn Wz is not maximal, there exists a W3 such that
WI E Wz E W3 E L. And so on.
Thus, when an ordinal belongs to a limit ordinal, a third party is
intercalated into the relation of belonging, and, as this process has
no stopping point, as there is no maximal element, it can be said that,
between any element W of a limit ordinal Land L itself, there is
always an 'infinity', in the intuitive sense, of intermediate ordinals.
So it is in a strong sense that the limit ordinal does not succeed. No
ordinal is the last to belong to it, the 'closest' to it. A limit ordinal is
always equally 'far' from all the ordinals that belong to it. Between
the element w of Land L, there is an infinite distance where inter-
mediaries swarm.
The result is that, contrary to what is the case for a successor ordinal,
a limit ordinal does not hollow out any empty space behind itself. No
matter how 'close' to L you imagine an element w to be, the space
between wand L is infinitely populated with ordinals. The limit ordinal
L is therefore in a relation of adherence to that which precedes it; an
infinity of ordinals 'cements' it in place, stops up every possible gap.
If the successor ordinal is the ontological and natural schema of
radical beginning, the limit ordinal is that of the insensible result, of
transformation without gaps, of infinite continuity. Which is to say
that every action, every will, is placed either under the sign of the
successor, or under the sign of the limit. Nature here furnishes us
with the ontological substructure of the old problem of revolution
(tabula rasa, empty space) and of reform (insensible, consensual and
painless gradations).
9.15. There is another way to indicate the difference between
successors and limits (which are for us the predicates of natural
The union of a set E is the set constituted by the elements of the
elements of E. This is related to a very important operator of the
ontology of the multiple, the operator of dissemination. The union
of E 'breaks open' the elements of E and collects all the products of
this breaking-open, all the elements contained in the elements whose
counting-for-one E assures.
An example: take our canonical example of three, the set T that
makes a triplet of the void, the singleton of the void and the pair of
the void and its singleton. It is written (0,(0),(0,(0))). What is the
union of T?
The first element of T is 0, which has no elements. It therefore
donates no elements to the union. The second element is (0), whose
single element is O. This latter element will feature in the union.
Finally the third element is (0,(0)), whose two elements are 0 (which
we already have) and (0). So in the end the union of T, the set of the
elements of its elements, is composed of 0 and (0): it is the pair (0,(0)).
That is to say, our D, or the number two. The dissemination of three
is no other than two. We state in passing (this will be clarified in
9.18) that the union of T is 'smaller' than T itself.
9.16. The position of ordinals with regard to union is most peculiar.
Given that an ordinal W is transitive, all its elements are also parts.
And this means that the elements of the elements of W, which are
also the elements of the parts of W, are themselves elements of W.
In the union of an ordinal we find nothing but the elements of that
ordinal. That is to say that the union of an ordinal is a part of the
ordinal. If we denote the set 'union of E' by uE, then, for every
ordinal, uW c W.
This property is characteristically natural: the internal homogene-
ity of an ordinal is such that dissemination, breaking open that which
it composes, never produces anything other than a part of itself. Dis-
semination, when it is applied to a natural multiple, delivers only a
'shard' of that multiple. Nature, stable and homogenous, can never
'escape' its proper constituents through dissemination. Or: in nature
there is no non-natural ground.
9.17. That the union of an ordinal should be a part of that ordinal,
or that the elements of its elements should be elements, brings us to
the question: are they all? Do we ultimately find not even a 'partial'
part (or proper part, compare 4.12), but only the ordinal we began
with? It could well be that every element can be found as element of
an element, since the internal fabric of an ordinal is entirely intri-
cated. In that case, uW = W. Not only would dissemination return
only natural materials, but it would restore the initial totality. The
dissemination of a natural set would be a tautological operation.
Which is to say that it would be absolutely in vain: we could then
conclude that nature does not allow itself to be disseminated.
9.18. This seductive thesis is verified in the case of limit ordinals, if
such a case exists.
Take any element WI whatsoever of a limit ordinal L. We have
shown (in 9.14) that between WI and L necessarily comes an inter-
calated element W2, in such a fashion that we always have (whatever
the element wd the chain WI E W2 E L. But, in addition, when we
disseminate L the element WI will be found again in the union, as an
element of W2. Consequently, every element of L features in uL, the
union of L. And, as we have seen, conversely (9.15), that every
element of uL is an element of L (since uL c L), it only remains to
conclude that the elements of L and those of uL are exactly the same.
Which is to say that L is identical to uL.
To dissemination, the limit ordinal opposes its infinite seIf-
coalescence. It is exemplarily natural, in so far as, in being 'dissected',
its elements do not alter. It is its own dissemination.
9.19. A successor ordinal, on the other hand, resists being identified
with its dissemination. It remains in excess of its union.
Let's consider a successor ordinal W. By definition it has a maximal
element WI' Now it is impossible that this element should be found
in the union of W. If it were found, that would mean that it was the
element of another element, Wz, of W: so WI E Wz, and WI would not
be maximal. The maximal element WI necessarily makes the differ-
ence between Wand uW. There is at least one element of a successor
ordinal that blocks the pure and simple disseminative restoration of
its multiple-being. A successor, unlike a limit, is 'contracted', altered,
by dissemination.
9.20. In my view, this contrast is of the greatest philosophical impor-
tance. The prevailing idea is that what happens 'at the limit' is more
complex, and also more obscure, than that which is in play in a suc-
cession, or in a simple 'one more step'. For a long time philosophical
speculation has fostered a sacralisation of the limit. What I have
called elsewhere I the 'suture' of philosophy to the poem rests largely
upon this sacralisation. The Heideggerian theme of the Open, of the
deposition of a closure, is the modern form of the assumption of
the limit as a wrenching away from counting, from technique, from
the succession of discoveries, from the seriality of Reason. There is
an aura of the limit, and an unbeing of succession. The 'heart come
from another age' aspires (and this horizon-effect is only captured,
so it seems, by the poem) to a movement across those 'infinite
meadows where all time stands still'.z
What the ontology of the multiple (based in a contemporary
Platonism) teaches us is, on the contrary, that the difficulty resides
in succession, and that there, also, resides resistance. Every true
test for thought originates in the localisable necessity of an addi-
tional step, of an unbroachable beginning, which is neither fused
through the infinite replenishment of that which precedes it, nor
identical to its dissemination. To understand and endure the test
of the additional step, such is the true necessity of time. The limit
is a recapitulation of that which composes it, its 'profundity' is
fallacious; it is in virtue of its having no gaps that the limit ordinal,
or any multiplicity 'at the limits', attracts the evocative and hollow
power of such a 'profundity'. The empty space of the successor
is more redoubtable, it is truly profound. There is nothing more to
think in the limit than in that which precedes it. But in the successor
there is a crossing. The audacity of thought is not to repeat 'to the
limit' that which is already entirely retained within the situation
which the limit limits; the audacity of thought consists in crossing a
space where nothing is given. We must learn once more how to
9.21. Basically what is difficult in the limit is not what it gives us to
think, but its existence. And what is difficult in succession is not its
existence (as soon as the void is guaranteed, it follows ineluctably)
but that which begins in thought with this existence.
And so, speaking of the limit ordinal, the question returns, ever
more insistent: do limit ordinals exist? On condition of the existence
of the void, there is 1, and 2, and 3 ... , all successors. But a limit
The reader will have realised: we find ourselves on the verge of the
decision on the infinite. No hope of proving the existence of a single
limit ordinal. We must make the great modern declaration: the infi-
nite exists, and, what is more, it exists in a wholly banal sense, being
neither revealed (religion), nor proved (mediaeval metaphysics), but
being simply decided, under the injunction of being, in the form of
number. All our preparations amount only to saying, to being able
to say, that the infinite can be thought in the form of number. We
know it, at least for that which falls within the natural ontological
horizon of number: the ordinals. That is infinite which, not being
void, meanwhile does not succeed. It is time to announce the
Axiom of Infinity. A limit ordinal exists.
Recurrence, or Induction
10.1. A momentary pause to begin with: let's recapitulate what the
ordinals give us to think as regards being qua being, from the view-
point of a philosophy informed by mathematical ontology.
10.2. The ordinals are, because of the internal stability of their
multiple-being (the maximal identity between belonging and inclu-
sion, between 'first' presentation through the multiple, as element,
and re-presentation through inclusion, as part) and the total homo-
geneity of their internal composition (every element of an ordinal is
an ordinal), the ontological schema of natural multiplicity.
10.3. The ordinals do not constitute a set: no multiple-form can
totalise them. There exist pure natural multiples, but Nature does not
exist. Or, in Lacanian terms: Nature is not-all, just as is being qua
being, since no set of all sets exists either.
lOA. The anchoring of the ordinals in being as such is twofold.
The absolutely initial point that assures the chain of ordinals of
its being is the empty set 0, decided axiomatically as secularised form,
or number-form, of Nothingness. This form is nothing other than
the situation-name of being qua being, the suture of every situa-
tion-being, and of every language, to their latent being. The empty
set being an ordinal, and therefore a natural multiple, we might say:
the point of being of every situation is natural. Materialism is founded
upon this statement.
10.5. The point-limit that 'restarts' the existence of the ordinals
beyond Greek number (the finite natural whole numbers; on Greek
number, see chapter 1) is the first infinite set, (0, decided axiomatically
as a secularised form - and thus entirely subtracted from the One - of
infinite multiplicity.
From this point of view, the ordinals represent the modern scale
of measurement (conforming to the two crucial decisions of modern
thought) of natural multiplicity. They say that nothingness is a form
of natural and numerable being, and that the infinite, far from being
retained in the One of a God, is omnipresent in nature, and, beyond
that, in every situation-being.
10.6. Our passage through the ordinals (or the limits of our repre-
sentation of them) arranges them according to an untotalisable
sequence. This sequence 'starts' with O. It continues through the
natural whole numbers (1,2, ... ,n,n + 1, ... , etc.), numbers whose
form of being is composed of the void (in the forms (0),(0,(0)),(0,(0),
(0,(0))), ... , etc.). It is continued by an infinite (re)commencement,
guaranteed by the axiom 'a limit ordinal exists', which authorises the
inscription, beyond the sequence of natural whole numbers, of (0, the
first infinite ordinal. This recommencement opens a new series of
successions: (0, (0 + 1, ... , (0 + n, ... , etc. This series is closed beyond
itself by a second limit ordinal, (0 + (0, which inaugurates a new series
of successions, and so on. Thus we achieve the representation of a
series of ordinals, deployed with no conceivable stopping point,
which transits within the infinite (beyond (0) just as in the finite.
10.7. The ordering principle of this sequence is in fact belonging
itself: given two ordinals WI and W
, then WI E W
, or W
or WI = W
• Belonging, a unique ontological relation because it
governs the thinking of multiple-being as such, is also that which
totally orders the series of ordinals. So that, if W is an ordinal and
S(W) its successor, then W E S(W). So that, if n is a natural whole
number (a finite ordinal) and n' a 'larger' whole number, then n E
n'. And so that, for any natural whole number n whatsoever, n E (0
(the first infinite ordinal), etc.
10.8. There are three types of ordinal (after the modern decisions
which impose the void and the infinite):
1 The empty set, 0, is the inaugural point of being.
2 The successor ordinals adjoin to their predecessor one element,
namely that predecessor itself. The successor of W is called S(W).
W is the maximal element in S(W), and the presence of a maximal
element allows us to characterise successors in a purely immanent
(non-serial) fashion. Successor ordinals give us a numerical schema
for what it means to say 'one more step'. This step consists always
in supplementing all that one has at one's disposal, with a unique
mark for that all. To take 'one more step' comes down to making
one of all of the given multiplicity, and adjoining that one to it.
The new situation is 'maximalised': it contains one term that
dominates all the others.
3 The limit ordinals have no maximal internal element. They mark
the beyond proper to a series without stopping point. They do
not succeed any particular ordinal, but it can be said that they
succeed all the ordinals of the sequence of which they are the
limit. No ordinal in this sequence is 'closer' to the limit ordinal
than any other. For a third ordinal, and ultimately an 'infinity'
(in the intuitive sense of a series with no stopping point) of ordi-
nals, will intercalate themselves (according to the order-relation,
which is belonging) between every ordinal of the sequence and
the limit ordinal. The limit ordinal adheres to everything that
precedes it. This is specifically indicated by its identity with its
own dissemination (L = uL). The limit totalises the sequence, but
does not distinguish any particular ordinal within it.
10.9. Just as a limit ordinal is structurally different from a successor
ordinal (as regards the internal maximum, and as regards dissemina-
tion), so the 'passage to the limit' is an operation of thought entirely
different from 'taking one more step'.
Succession is, in general, a more difficult local operation than the
global operation of passage to the limit. Succession gives us more to
think about than does the limit. The widespread view to the contrary
stems from the fact that, not being 'absolutely modern', we still tend
to sacralise the infinite and the limit, which is to say: retain them still
in the form of the One. A secularised thought, subtracted from the
One and the sacred, recognises that the most redoubtable problems
are local problems, problems of the type: 'How to succeed?', 'How
to take one more step?'.
10.10. The space of the ordinals allows us to define the infinite and
the finite. An ordinal is finite if, in the chain of order governed by
belonging, it comes before 0). It is infinite if it comes after 0) (includ-
ing 0) itself).
We will find that, just as Dedekind's intuition suggested, only the
existence of an infinite ordinal permits us to define the finite. Modern
thought says that the first situation, the banal situation, is the infinite.
The finite is a secondary situation, very special, very singular,
extremely rare. The obsession with 'finitude' is a remnant of the
tyranny of the sacred. The 'death of God' does not deliver us to fini-
tude, but to the omnipresent infinitude of situations, and, correla-
tively, to the infinity of the thinkable.
10.11. The final synthetic recapitulation of the fact that the ordinals
give us to think being qua being, in its natural proposition, is
complete. Now we must turn towards our capacity to traverse and
to master rationally this donation of being. One way to do so is
simply to proceed, in this boundless fabric, to the carving-out of
10.12. It is a blessing for our subjective finitude that the authority-
properly without measure - of natural multiplicities allows that diag-
onal of passage, or of judgement, which is reasoning by recurrence,
also called complete induction and, in the case of infinite ordinals,
'transfinite induction'. In fact this alone allows us, in treating of an
infinite domain (and even, if we consider the ordinals, one that is
infinitely infinite), to anticipate the mome1tt of conclusion.
Suppose that we wish to show that all ordinals possess a certain
property P. Or that we wish to establish rationally, with a proof, a
universal statement of the type: 'For all x, if x is an ordinal, then
P(x),. How can this be achieved? It is certainly impossible to confirm
case by case that it is so: the task would be infinitely infinite. Neither
is it possible to consider the 'set of ordinals', since such a set does
not exist. The 'all of the ordinals', implied in the universal quantifier
of the statement 'for all x', cannot be converted into 'all the elements
belonging to the set of ordinals'. Such a set is inconsistent (see 6.11).
It is precisely the alleviation of this impasse that is the business of
reasoning by recurrence.
10.13. Reasoning by recurrence combines one verification and the
demonstration of one implication. Once in possession of these two
moments, the structure proper to the ordinals authorises the universal
Take property P. We begin by confirming that the empty set 0 pos-
sesses this property; we test P for the 'case' of O. If the empty set does
not possess the property P, it is pointless to pursue the investigation.
If one ordinal, 0, does not have property P, it is certainly false that
all ordinals do. Suppose, then, that the statement P(O) is true, that
the test in the case of 0 is positive.
We will now try to prove the following implication: if all the
ordinals that precede some ordinal W (according to the total ordering
of the ordinals, which is belonging) have the property P, then W also
has it.
Note that this implication does not tell us that an ordinal with
property P exists. It remains in the hypothetical register, according
to the general pattern: 'if x is so, then what follows x is so'. The
implication is really universal, it does not specify any ordinal W. It
says only that, for every ordinal W, supposing that those which
precede it in the chain of ordinals satisfy P, one is compelled to admit
that W satisfies it also.
It is usually necessary to divide this demonstration (supposing that
it is possible, which obviously depends on property P), by treating
the case where we suppose W to be a successor separately from the
case where we suppose it to be a limit (since W is any ordinal what-
soever, it could be one or the other). Reasoning by recurrence, as we
saw in the central implication that constitutes it, strongly binds that
which is the case for an ordinal W to that which is that case for the
ordinals that precede it. Now the relationship of a limit ordinal to
the anterior ordinals (one of infinite adherence) differs radically from
that of a successor (which, between itself and its predecessor, clears
an empty space). Because of this, the procedures of thought and of
proof put into play in the two cases are usually heterogeneous. And,
as we might expect, given the philosophy of this heterogeneity (d.
9.19), it is generally the case of the successor that is the most
Assume that we have verified the truth of P(O), and that we have
proved the implication 'if, for every ordinal w that precedes W (that
belongs to W: order is belonging), it is the case that P(w), then it is
also the case that P(W)'. We can conclude that all ordinals satisfy P,
in spite of the fact that this 'all' not only alludes to an infinitely infi-
nite immensity of multiples, but that, even so, it does not make an
All. It is truly the infinite and inconsistency 'conquered word by
10.14. What authorises such a passage to 'all', such an ambitious
'moment of conclusion'? The authorisation is granted us by a funda-
mental property of the ordinals as ontological schema of the natural
multiple: their 'atomistic' character, the existence, for every property
P, of a minimal support for this property as soon as one ordinal pos-
sesses it. (See 8.10 and 8.15).
If the conclusion were false - if it were not the case that all ordinals
possess property P - that would mean that there was at least one
ordinal which did not possess property P. This ordinal would then
possess the property not-P, not-P meaning simply 'not possessing
property P, being a non-P'.
But, if there exists an ordinal that possesses property not-P, there
exists a smallest ordinal which possesses this property 1ZOt-P, by
virtue of the atomistic principle, the principle of minimality. And,
since it is the smallest to possess property not-P, all those which are
smaller than it must possess property P.
We could object: these ordinals 'smaller than it' may not exist,
because it is possible that the minimal ordinal for the property not-P
is the void, which is not preceded by anything. But no: since (first
moment of our procedure) we have verified precisely that 0 possesses
the property P, the minimal ordinal for not-P cannot be o. Thus it
does make sense to speak of ordinals smaller than it; they exist, and
must all possess property P.
Now our central implication, supposed proved, said exactly that,
if all the ordinals smaller than a given ordinal possess property P,
then that ordinal also possesses it. We have reached a formal contra-
diction: that the supposed minimal not-P must be a P. It is necessary
then to conclude that this latter does not exist and that therefore all
ordinals do possess property P.
Thus the ontological substructure of natural mutiplicities comes
to found the legitimacy of recurrence. Our verification (the case of
0) and our demonstration (if P(w) for all w such that w E W, then
P(W) also), if it is possible (which depends on P ... and on our math-
ematical know-how), authorises the conclusion for 'all ordinals'.
10.15. We have remarked, in studying Peano's axiomatic (see 5.3)
that reasoning by recurrence is a fundamental given of serial numeri-
cality, of which the natural whole numbers are an example. It is quite
natural that it should extend to that 'universal series' composed by
the ordinals. But the great difference is that, whereas in Peano the
principle of induction or recurrence is an axiomatic form or a formal
disposition, here, since it is founded in being (in the theory of the
pure multiple), it is a theorem - that is, a property deducible from
the ordinals.
It is of the essence of the natural multiple, which escapes all total is-
ing thought, to submit itself nonetheless to that intellectual 'capture'
which is the inductive schema. Here, once more, being is found to be
amenable to thought in that form of Number which is the conclusion
for 'all', proceeding both from the verification for one only (here, 0)
and from a general procedure which transfers the property of what
comes 'before' (predecessor or endless series, depending on whether
it is a case of a successor ordinal or a limit ordinal) to what comes
'after'. Number is that which accords being to thought, in spite of
the irremediable excess of the former over the latter.
10.16. Reasoning by recurrence is a proof-procedure for universal
statements concerning ordinals. It allows us to conclude. But there is
a more important usage of recurrence, or of transfinite induction, one
which allows us to attain the concept. This is inductive definition.
Suppose that the aim of our thinking is not to prove that this or
that type of multiple, for example ordinals, has property P, but to
define property P in a way that would allow us then to test it on
multiples. A well-known difficulty in such a case is that we don't
know in advance whether a property defined in language is 'applica-
ble' to a pure multiple without inconsistency resulting. We have seen,
for example (in 2.11), that the property 'not being an element of
itself' does not apply to any existing set, and that its perfect formal
correctness does not alter the fact that, handled without care, it leads
to the ruin, by way of inconsistency, of all formal thought. But how
can we introduce limitations and guarantees, if language alone cannot
support them? The procedure of definition by recurrence, or inductive
definition, answers this question.
10.17. What will found the legitimacy of the procedure this time is
the fact that, with the ordinals, we have at our disposal a sort of
universal scale, which allows us to define property P at successive
levels, without exposing ourselves to that danger of inconsistency that
attends on any supposition of an All. Inductive definition is a ramifi-
cation of the concept: property P will not be defined 'in general', but
always as indexed to a certain level, and the operators of this index-
ation will be the ordinals. Here, once again, being comes to the aid
of finitude, in assuring for our thought, which the domain of being
as pure multiple exceeds on all sides, that it can proceed in steps, in
10.18. In conformity with the typology of ordinals, which distin-
guishes three types (the void, successors, limits), our procedure is
divided into three.
1 We first define explicitly, with a statement, level 0 of the pro-
perty. An explicit definition assumes that we have a property -
say, Q - already defined, and that we can affirm that level 0 of
P - say Po - is equivalent to Q. We would then have: Po(x) H
2 We then say that, if level w of P is defined, P
, then level S(w),
that is, PS(w)' is defined through an explicit procedure to be indi-
cated. To say that P
is defined is to say that there is a property
- call it R - already defined such that P w is equivalent to it, so
Pw(x) H R(x). The existence of an explicit procedure enabling us
to pass from the definition of P w to that of PS(wl means that there
is a function f that assures the passage of R (which defines P
) to
a property f(R) which will define PS(w)' Finally, we can say that 'x
has the property PS(w)' means 'x has the property f(R)', or that f,
which permits the 'passage' from the definition of P w to that of
PS(w)' is an explicit operation on R, fixed once and for all.
3 Finally, we will say that, if all the levels of P below a limit ordinal
L have been defined, say: PO,P
••• ,Pn,P
••• , then level L of
P, say for example P
, is defined by a 'recollection' that can be
explicated by that which defines all the levels anterior to it (in
this process, union or dissemination generally plays a decisive
role, for reasons given in 9.17). Usually we have something like:
for a given x, Pdx) is true, if there exists a level below L, call it
w, where w E L, for which P w(x) is true. The limit level, in con-
formity with its essence, will assume all the inferior levels and
will not introduce anything new.
Thus we will have at our disposal not just a single concept P, but an
infinite and infinitely ramified family of concepts, from Po, explicitly
defined, up to the more considerable ordinal indexations P w, passing
through Pn,PO)'P
, etc. We will then be able to say that concept P, as
unique concept, is defined by transfinite induction, in the following
sense: for a given x, P(x) will be true if and only if there exists an ordinal
W such that x possesses the property at level W. We would have the
following equivalence: P(x) H 'there exists a W such that Pw(x)'.
So the inductive mastery of the concept passes by way of its ordinal
ramification, and by way of the equivalence between 'the concept P
holds for x' and 'the concept P holds for x at level W of that concept'.
This equivalence avoids all mentioning of the All. It tests the property
P not 'in general', but on one level, thus freeing it from paradoxes of
10.19. I shall give a most interesting example; its interest is both
intrinsic (it sheds a keen light on the general structure of the theory
of the pure multiple, or ontology: it proves that, thought in their
being qua being, multiples are stratified) and methodological (we will
see clearly the functioning of levels in the definition of the concept).
The underlying idea is to define, for each multiple, an ontological
rank, indexed on the ordinals, which measures its 'distance', in a
certain sense, from that initial suture which is the empty set. We
could also say that the rank is a measure of the complexity of a
set, of the immanent intrication of the instances of the void that
constitute it.
Naturally, it is impossible to speak of 'all' sets: to do that it would
be necessary to collect them as the elements of a set of all sets, which
would be inconsistent. The prudent, gradual approach of the induc-
tive procedure is indispensable here.
The two important operations of set theory which allow one to
'step' from one set to another are:
1 Union, or the set of elements of elements of the initial set; the
operation of dissemination, which we have already met (compare
9.15). Given a set E, we denote its union by uE.
2 The set of parts, which consists of 'gathering together' to make
one all the parts of the initial set, all that is included in that set
(on belonging and inclusion, see 7.3). We denote by pIE) the set
of the parts of E. Note that the elements of pIE) are the parts of
E: if e E PIE), then e c E.
We will construct the hierarchy of ranks by means of these two
operations. The property we will try to define through transfinite
induction, according to the method explained in 10.18, will be
denoted by R(x), to be read as: 'x possesses a rank' (or: 'x is well-
founded'). Our three steps will be as follows:
1 Explicit definition of the property at lellel O. We propose that
Ro(x) is not true for any x, in other words that Ro(x) is equivalent
to x E O.
2 Uniform treatment of successive levels. We posit that Rs(w)(x) is
true if and only if x belongs to the set of parts of the set consti-
tuted by all the z which satisfy Rw. In other words, the rank at
successor level S(w) is the set of parts of the rank defined for the
level which the predecessor w indexes. This can be written as
follows: Rs(w)(x) H ((y E x) Rw(Y)): if x satisfies Rs(w), the ele-
ments of x satisfy R
, and consequently x is a part of the set of
sets which satisfy Rw. We could also write, denoting by Rw the
set of x for which Rw(x) is true:
3 Uniform treatment of limit levels. As would be expected, it is
union that is at work here. We will say that Rdx) is true if x is
of a rank whose index is smaller than L, that is, if there exists
aWE L for which Rw(x) is true. Thus the rank RL recollects all
the elements of the ranks below it; it is the union of these ranks.
With the same conventions as above, we can write: (x E Rd
H x E uRw for all W smaller than L.
Property R is thereby totally defined by induction. We will say that
x possesses a rank, or that R(x) (without index) is true, if a (successor
or limit) ordinal W exists for which Rw(x) is true. This property
'means' that one arrives at the complexity of x, beginning from 0
(which defines level Ro of the property), through the successive
employment of union and of passage to the parts, an employment
whose 'length' is measured by an ordinal: the smallest ordinal w for
which Rw(x) is true.
10.20. That this procedure really 'works', that it makes sense ulti-
mately to speak of the property R, however, is not self-evident. The
generosity of natural being consists in the fact that one can prove the
effectivity of this ramified determination of the concept.'
Thus thought proceeds in its passage through being, under the
universally intricated and hierarchised rule of Nature, which doesn't
exist, but provides measurable steps. Number is accessible to us
through the law of such a passage, at the same time as it sets the
conditions - as we saw with the ordinals - for this passage itself.
Number is that through which being organises thought.
Natural Whole Numbers
11.1. The ordinals directly give us the Greek numbers: natural whole
numbers. We are even in a position to attach a new, non-Greek,
legitimacy to the adjective 'natural' which mathematicians, with the
symptomatic subtlety of their nominations, adjoin to the civil status
of these numbers: they are 'naturals' by virtue of the fact that, within
the realm of the finite, they coincide purely and simply with the
ordinals, which constitute the ontological schema of the pure natural
For it is 'natural' to identify, in its being, the place of number (that
is, of whole number) - a place whose existence Dedekind vainly tried
to secure on the basis of the consideration of 'all the possible objects
of my thought' - with the first infinite ordinal 00, whose existence we
decide, under the modern injunction of being, as we enounce the
axiom 'a limit ordinal exists'.
11.2. To say that 00 is the place of whole number has a precise set-
theoretical meaning: what 'occupies' the place is that which belongs
to it. Now, not only do all ordinals that precede a given ordinal
belong to it; they constitute all the elements of that initial ordinal.
In fact, we know that total order over the ordinals is really belong-
ing (see. 8.10). And, consequently, an ordinal smaller than a given
ordinal W is precisely an ordinal that belongs to W. The image of an
ordinal (for example, one larger than (0) is as follows:
Oe le2e ... enen+le ... eooeoo+le ... eW
where all the numbers in the chain of belonging constitute precisely
the elements of W. Visualised like this, the ordinal W appears as a
sequence of 'embedded' ordinals, whose 'length' is exactly W. There
are W links in the chain in order to arrive at W. We might also see
an ordinal W, containing exactly W ordinals (all those that precede
it), as the number of that of which it is the name. Which is another
way of saying that it is identified with the place where its predecessors
insist, being the recollection of that insistence.
Thus the definition of natural whole numbers is entirely limpid:
an ordinal is a natural whole number if it is an element of the
first limit ordinal co. In which case, the structure of the place of
number is:
Oe le2e ... enenen+le ... eco
But we must take care to note that co itself, which is the name of
the place, is not a part of it, since no set belongs to itself (d. 8.14).
The place of whole number, co, is not an element of that place, it is
not a whole number. As CO is the {irst limit ordinal, it follows that all
whole numbers, except the empty set 0 of course, are successors.
11.3. An attentive reader might object as follows: I say that co is the
first limit ordinal. But am I sure that a 'first' limit ordinal exists? The
Axiom of Infinity (9.20) says only: 'a limit ordinal exists', it does not
specify that this ordinal is 'the first'. What authorises us to call co the
'first limit ordinal', or first infinite ordinal? It could well be that, as
soon as I announce that 'a limit ordinal exists', a multitude of them
appear, none of which is 'first'. There could be an infinite descending
chain of such ordinals, just like the descending chain of negative
numbers which, it is clear, has no first term: no whole negative
number is 'the smallest', just as no whole positive number is 'the
largest' (this second point in fact comes back to saying that co, the
beyond and the place of the series of positive numbers, is a limit
But if I cannot unequivocally determine and fix the first limit
ordinal, then what becomes of my definition of whole numbers?
11.4. We can overcome this objection, once more, thanks to that
great principle of natural multiples that is minimality. We know that,
given a property P, if an ordinal exists that satisfies that property,
then there is one and only one minimal ordinal that satisfies it. Take
the property 'being a limit ordinal'. There certainly exists an ordinal
that satisfies it, since the Axiom of the Infinite says precisely that.
Thus, there exists one and only one limit ordinal that is minimal for
this property. Consequently we can speak without hesitation of a 'first
limit ordinal', or of the 'smallest limit ordinal', and it is to this unique
ordinal that we give the proper name 00. There is therefore no ambi-
guity in our definition of natural whole numbers.
11.5. We must never lose sight of the fact that notations of the type
t,2,n, etc. are ciphers, in the sense of codes, which serve to designate
multiples fabricated from the void alone. We have known for a long
time (already in 8.3) that t is in reality the singleton of the void, that
is, (0), that two is the pair of the void and the singleton of the void,
that is, (0,(0)), that three is the triplet of the void, the singleton of
the void, and the pair of the void and singleton of the void, that is,
(0,(0),(0,(0))), etc. To exhibit further this weaving of the void with
itself, let's also write down the real being of the cipher 4:
(0,(0),(0,(0) ),(0,( 0),(0,(0))).
Evidently 4 is a set of four elements, in the order 0, then (0), then
(0,(0)), then (0,(0),(0,(0))). These four elements are none other than
zero, 1,2 and 3. The elements of a whole number comprise precisely
all those numbers that precede it, which is not surprising, since we
have shown above that this is the innermost structure of every ordinal
(11.2). We could write: 4 = (0,1,2,3). And, as we have said, to pass
from 3 to 4 (as from any n to n + 1), we 'adjoin' to the elements of
3 (or of n) the number 3 itself (or the number n). Which is not sur-
prising, since this is the general definition of succession in the ordinals
It would obviously be impossible to use the procedure of succes-
sion to 'step' from some whole n, no matter how large, to the first
limit ordinal 00. This is because 00, let us repeat, is not a whole
number, it is the place of such numbers. An important law of thought
emerges here (one which, we might say in passing, the Hegelian figure
of Absolute Knowledge, supposed to be the 'last' figure of Conscious-
ness, contravenes), which states that the place of succession does not
itself succeed.
11.6. Once we have at our disposal the place of natural whole
numbers, their multiple-being which weaves the void through the
finite, and the law of succession as law of our passage through these
numbers, we 'rediscover' the classical operations (addition and mul-
tiplication for example) through simple technical manipulations that
arise from the general principles of inductive definition, or definition
by recurrence, explained and legitimated on the basis of natural being
in chapter 10. It is time to give a new example.
11.7. Take a given number, say tor example 4. We want to define
through induction a function F whose outcome will be as follows:
for any number n whatsoever (therefore for every whole number, and
there is an infinity of them), F(n) is equal to the sum 4 + n. To achieve
this, we have at our disposal only one operator: ordinal succession,
since the only thing we know is that all the whole numbers except 0
are successors. We will proceed exactly according to the schema
explained in 10.18, except that we will not have to worry ourselves
about the case of limit ordinals (since there are none before (0). We
will, as before, use S(n) to denote the successor ordinal of the whole
number n.
1 We will first state: F(O) = 4 (an explicitly given value, the underly-
ing intuition being that 4 + 0 = 4).
2 Then we will proceed to the successional induction by positing:
F(S(n)) = S(F(n)). A regular and uniform relation between the
value of the function for S(n) and its value for n, a relation that
uses only what we already know; the operation of succession,
defined in general on the ordinals. The underlying intuition is that
4 + (n + 1) = (4 + n) + 1, to return to the usual 'calculating' nota-
tion where the successor of n is denoted by n + 1.
The value of the function is defined entirely by these two equa-
tions. Say, for example, that I wanted to calculate F(2). I would have
the following mechanical sequence:
We can see clearly that such a schema is a true definition of addi-
tion, through the use of recurrence, on the basis of the operation of
succession alone. Once we have obtained this general inductive
schema of addition, multiplication can be similarly defined. Take the
function to be defined, P(n), whose value is n multiplied by 4. We
begin the induction this time with 1 and not with 0, stating that if
F(n) is as above (defining 4 + n inductively):
P(1) = 4 (guiding intuition: 4 x 1 = 4)
P(S(n)) = F(P(n)) (guiding intuition: 4 x (n + 1) = 4 + (4 x n))
These technical manoeuvres are of no direct interest. They serve only
to convince us that whole numbers thought in their being (ordinals
that precede 00, fabricated from finite combinations of the void) are
indeed also the same ones with which we count and recount without
respite, as the epoch prescribes us to do.
11.8. The philosophico-mathematical reconstruction of whole
numbers is now complete. They do not derive from the concept
(Frege), nor can their place be inferred from our possible thoughts
(Dedekind), nor is their law limited to that of an arbitrarily axioma-
tised operational field (Peano). They are, rather, in the retroaction of
a decision on the infinite, that part of number which being provides
to us in its natural and finite figure.
The whole numbers are Nature itself, in so far as it is exposed to
thought only to the limited extent of its capacity for finitude. Again,
this exposition is possible only on condition of a point of infinity, the
limit ordinal 00, the existential guarantee of whole number. This point
of infinity is immense in relation to the whole numbers, since, sub-
tracted from successoral repetition, it constitutes the place of their
total exercise, a place without internal limits (succession can always
continue). Nevertheless, it is minute in relation to the profusion of
natural infinite being beyond its first term 00. Whole number is the
form of being of the finite 'almost nothing' deployed by being qua
being between the void and the first infinity.
11.9. It is only in an anticipation without solid foundation, and in
homage to their antiquity, that we call the natural whole numbers
'numbers'. We have already remarked (8.8) that, still without a
general concept of number at our disposal, it would be illegitimate
to say that the ordinals were numbers. Now, the whole numbers are
none other than the ordinals. And number, or rather Number, quali-
fies a type of being of the pure multiple which exceeds the ordinals.
Until we have made sense of this type, in such a way that it becomes
applicable to all species of number (whole, relative, rational, real,
ordinal, cardinal), we can only speak of 'number' in a sense still
insufficiently liberated from its operational intuition, or from the
historical heredity of this signifier.
But our preparations are complete. The homage paid to the Greek
numbers is only the last act of a vast introduction, genealogical and
then conceptual. Now it is necessary to define Number.
Ontology of Number:
Definition, Order, Cuts, Types
The Concept of Number:
An Evental Nomination
12.1. The first part of this book was historical and critical (a study
of the great enterprises of the past). The second was constructive and
conceptual (the determination of the ordinals as schema of natural
multiplicity, on the basis of the concept of transitive sets). In this
third part, we are going to proceed regressively, axiomatically: we
shall begin with a general definition of Number, a remarkably simple
definition involving only the concept 'ordinal'. Then, by way of
increasingly specific determinations, we shall address the essential
attributes of the resulting concept of Number: total order, the process
of cutting, and finally - in the last place only - operations. In so
doing, we shall demonstrate how all of our traditional numbers (the
wholes, the rationals, the reals, and the ordinals themselves, con-
ceived and handled as Numbers) are only particular cases of the
general concept.
In my view, the three most important aspects of these proceedings
are as follows:
1 Considerations of order and operations arise from the intrinsic,
or ontological, definition of Number. Number is therefore not
itself an operational concept, it is a particular figure of the pure
multiple, which can be thought in a structural and immanent
fashion. The operational dimensions are only subsequent
traits. Number is not constructed; on the contrary, its very
being makes possible all of the constructions in which we
engage it.
2 The ordinals constitute the base material for the definition of
Number, its natural ontological horizon. But, taken in all their
generality, Numbers are 'non-natural' deductions from this natural
3 Our traditional numbers are only very specific cases, which cer-
tainly fall under the general and unified concept of Number but
by no means exhaust it. There remains an innumerable immensity
of Numbers we have not yet thought or used.
12.2. Definition: A Number is the conjoint givenness of an ordinal
and a part of that ordinal.
A Number will be denoted by the letter N, followed by indices to
distinguish between several different Numbers.
In other words, a number N is constituted by:
- an ordinal W;
- a subset F included in this ordinal, such that FeW.
The ordinal will be called the matter of Number, which we will
denote by M(N).
The part of the ordinal will be called the form of the Number,
which we will denote by F(N).
That part of the matter which is not in the form, that is, those ele-
ments of the ordinal W which are not in the part F(N), constitute the
residue of the Number. We denote this by R(N). The residue is equal
to the matter minus the form, and therefore to the set M(N) - F(N).
It is clear that, if we add together the form and the residue, we
end up with the whole of the matter. Therefore, using u to stand for
union (see 9.15): F(N) u R(N) = M(N).
Since a Number is entirely determined by its matter (an ordinal)
and its form (a part of that ordinal), it will often be convenient to
write it as a pair (M(N), F(N)), with the convention that the ordinal-
matter is written to the left, and the form to the right.
12.3. In examining this definition, the reader must observe a number
of precautions.
(1) We are dealing with a pure definition, a priori for the moment:
it is of no use, nor is it possible, to try to 'recognise' straightaway, in
this definition, any of our familiar numbers.
I will give an example: take as matter the ordinal 1 (whose onto-
logical composition is (0), the singleton of the void), and, as form,
the ordinal 0, the void, which is of course a part of 1, as it is a part
of every set (7.9). Using the above convention, we have the Number
N = (1,0). All we know is that, according to the definition (the result
of an ordinal and a part of that ordinal), N is a Number. The signs
'1' and '0' do not directly refer to any Number, since we have not
yet even established that we are dealing with Numbers. In fact, what
these signs 1 and 0 are going to indicate here - each on its own
account, a matter and a form - cannot be understood as Numbers,
since we do not discern in their writing the fundamental dual given-
ness of all Number: a matter and a form. A Number must involve
two marks, that of its matter and that of its form: now '1' is only
one mark, as is '0'. It would therefore be illusory to 'recognise' in the
Number (1,0) any familiar number whatsoever, on the pretext that
one 'recognises' 1 or O. At the moment, we have in (1,0) only an
abstract example of a Number, conforming to the concept of Number
given in the definition.
Just as the prisoners in Plato's cave once again make the descent
from the Idea (of Number) back down to the empirical (numbers),
we will demonstrate much later, in chapter 16, that the Number (1,0)
is the true concept of the familiar negative number -1. But at this
stage it is essential that the reader consider the examples as simple
clarifications of the definition of Number, and not seek to reconnect
them to the cavernous empirical domain of numbers.
(2) The matter of a Number is an ordinal; we have said enough about
ordinals for there to be no mystery about this. On the other hand, the
form of a particular Number is constrained only to be a part of that
ordinal, a set included in the ordinal. The general concept of a part (or
subset) is somewhat indeterminate and, when the matter happens to
be infinite, offers no foothold for intuition. In particular, note:
- that this part might be empty (compare the example above);
- that this part might be the entire ordinal; if we take for matter
the limit ordinal 00, and for form this same ordinal (which is a
'total part' of itself), we obtain a wholly permissible Number
(conforming to the definition), which is written N = (00, (0); we
will see in chapter 16 that there are excellent reasons to allow
that this Number is none other than the ordinal 00 itself, but at
the moment this is not at all obvious;
- that this part does not necessarily have to be contained or con-
nected as one; it could be dispersed, lacunary, composed of scat-
tered elements, and so on; for example, if we take as matter the
limit ordinal 00, we can take as form the set constituted by the
whole numbers 3, 587 and 1165. These three finite ordinals are
all elements of 00, and therefore, taken together, they form a part
ofoo. We will have a quite permissiblenumberN =(00,(3,587,1165)),
whose form has three completely separate elements.
12.4. These formal possibilities make a visualisation of Number dif-
ficult. We can imagine spatial designs somehow like this:
I) Number whose form is connected
3) Number whose form is void
2) Number whose form is dispersed
4) Number whose form takes up the
whole of the mailer
But doubtless the simplest way is to have recourse to a linear
arrangement (see below). This figuration is based on ordinal linearity,
conceived as a universal series from which the being of Number is
A line segment, whose supposed origin is the ordinal 0, represents
the 'ordinal axis'. We mark with an asterisk"" upon this axis the
matter of the Number, an ordinal W. We mark with an emboldening
of the line the form of the Number, part of its matter. The rest, left
unchanged, represents the residue. If we want to represent a particu-
lar ordinal, we can do this with a little circle on the arrow, with the
name above or below. With these conventions, a Number will look
like this:
o form W = ordinal-matter
. ~ - - - - ~ - - - - ~ *
residue ordinals
Once again, this type of drawing can aid comprehension, but can
also be an encumbrance. Its principal failing, which it shares with the
famous 'Venn diagrams' used to teach schoolchildren operations on
sets (union, intersection, etc.), is that it habituates one to imagining
that a part of a set is a sort of continuous whole, a compact neigh-
bourhood. Now the sole prerequisite of a part is that it should
contain only elements of the set of which it is a part. These elements
might very well be highly dispersed, scattered to the far regions of
the initial set, and the visual schema of a part, to indicate this disper-
sion, must be able to be punctured, fragmented, dismembered. The
unfortunate thing is that the drawing then loses any intuitive value
it might have had: one simply gets the impression that there are many
parts. In looking at my lines and their emboldenings, one must always
keep in view, conceptually, that there is no reason for the form of
a Number to be a continuous segment, but that it could well be
dispersed throughout the full extent of the ordinal-matter, as could
the residue.
For example, the Number mentioned above, which has for matter
the limit ordinal (0 and for form the triplet (3,587,1165), must be
represented somehow like this (with the additional complication that
the infinity of (0 is not truly 'commensurable' in a drawing):
o 3
• •
• form
1165 ro
12.5 The following section is entirely dedicated to a philosophical
elucidation of our definition.
We will begin with the capital N with which I furnish Number.
In all attempts undertaken to determine the concept of number,
the problems of terminology bring the weight of the event to bear
upon the researcher.
Take for example the appellation 'irrational numbers'. It is truly
astonishing to find such a designation at the heart of mathematical
rationality. The doctrine of 'cuts' forged by Dedekind is nothing other
than the determination - wholly rational and demonstrative - of the
concept of irrational number. But exactly the same could be said for
the theory of proportions in Euclid's Elements. It is clear, then, that
'irrational', in these mathematical texts whose rationality is transpar-
ent, paradigmatic even, no longer has any signification.
We might say that what makes itself known here is a symptom of
the radical difference between nomination and signification. A signi-
fication is always distributed through the language of a situation, the
language of established and transmitted know ledges. A nomination,
on the other hand, emerges from the very inability of signification to
fix an event, to decide upon its occurrence, at the moment when this
event - which supplements the situation with an incalculable hazard
- is on the edge of its disappearance. A nomination is a 'poetic' inven-
tion, a new signifier, which affixes to language that for which nothing
can prepare it. A nomination, once the event that sustains it is gone
forever, remains, in the void of significations.
Now, at the moment of the great Greek crisis of number, when
the arrival of that at once inevitable and enigmatic event made it
known that certain relationships (those, for example, of the diagonal
of a square and its side) cannot be 'numbered' within the code of
existing numbers, the word alogos arrived, saturating and exceeding
the mathematical situation. This word designates that which, having
no logos, nonetheless must be decided as number. It inscribes in a
new situation of thought a nomination without signification: that of
a number which is not a number.
Since that time, the word has lodged itself, without alteration,
in mathematical language. It traverses translations, negligible but
subsistent. Our word 'irrational' is unmindful of the import of
the nomination alogos to the same extent that the word 'rational'
retains little of the Greek logos. And, above all, this nomination
has ended up taking on a univocal signification. But the contrast
remains, and one can reactivate it - as I do - in between signification
and that which, in the word that imparts it, contradicts it explicitly.
For this contrast is the trace within language of a foundational
truth -event.
It can easily be shown that the same applies for 'real' numbers,
or for 'imaginary' numbers. Even Cantor's reason for calling the
ordinals beginning with ro 'transfinite' numbers becomes less and less
obscure for us now, connected as it is to his mindfulness of offending
the sanctity of the Infinite with his invention.
The frequency in number-theory of a gap between the trace of a
nomination and the sediments of signification indicates that the think-
ing of number is a true evental site: it represents in mathematics a
zone of singular precarity and sensitivity, struck regularly by the
excess of an event that language and established know ledges consider
destitute of signification, and whose destiny can oilly be sustained by
means of a poetic and supernumerary nomination.
And this is because number is, amongst the forms of being, that
one which opens onto our thought by way of its organisation (see
10.20). Which means that everything excessive that thought encoun-
ters in number, everything that interrupts the regime of its being
by way of an evental caesura, has immediate disorganising effects
for thought.
12.6. My doctrine of Number, even if my terminology and the echo
I give it in philosophical thought are very different things, is neverthe-
less substantially that of 'surreal numbers' invented by J. H. Conway
in the seventies (see 1.7).1 I make no claim at all to having produced
anything new of a strictly mathematical order. Why, then, change
'surreal number' to just 'Number', with a capital N?
It is basically a poetical disagreement. The nomination proposed
by Conway seems to me rather too narrow; let's say that it belongs
to an oneiric genre ('surreal' obviously suggesting 'surrealist'), whereas
the excessive nature of the discovery in my view demands the majestic
genre of the epic, something capable of conveying the unanticipated
royal arrival of Number as such.
More technically, it seems to me that 'surreal' remains caught
within the notion - all too highly charged with meanings - of a con-
tinuity through successive widenings. The adjective 'surreal' seems to
suggest itself because these new numbers 'contain' the real numbers
(as they contain the ordinals); as if the new space conquered was an
extension of the old. In his book, Gonshor (see 1.7), seeking to make
propaganda for the surreals, declares that 'we now know the exciting
fact that the surreals form a field containing both the reals and the
ordinals.'2 But what is exciting in the discovery, at least for the phi-
losopher, goes well beyond this algebraic collection of reals and
ordinals. It relates rather to a complete reinterpretation of the very
idea of number, to the possibility of finally thinking number as a
unified figure of multiple-being. That reals and ordinals arise within
this figure is the least of the matter, a simple consequence. And all
the more so given that, along with reals and ordinals, the misnamed
'surreals' contain an infinitely infinite throng of numbers whose exis-
tence no one has conceived of before, and which retroactively make
our historical numbers seem like a miniscule deduction from all those
abundant varieties of numerical being. To give just one example:
surreal numbers permit a complete doctrine not only of infinitesimal
numbers, but of an infinity of infinitely small numbers, describing a
'downwards' numerical swarming just as vast as that which the ordi-
nals describe 'upwards'.
To use a political image: the nomination 'surreal' seems to me to
be marked by that caution, by that attachment to old significations,
that characterises a certain 'reformist' reserve when confronted with
the event. Now, I think - I wager - that we must adopt the language
of rupture here, the 'revolutionary' language. I will say therefore that
what takes place here is nothing less than the advent to our thought
of Number.
Ultimately the capitalisation of Number does not so much distin-
guish the genera from the species subsumed to it (whole numbers,
rational numbers, real numbers, ordinal numbers, infinitesimal
numbers, etc.) - although it does indeed activate such a distinction-
as it emphasises the gap between a nomination (here at last is Number)
and the diverse significations that, having once been nominations
themselves, have become the names of numbers.
12.7. Making thus our wager on the word Number, let us try to
legitimise the definition: 'A Number is constituted by the conjoint
givenness of an ordinal and a part of that ordinal.'
The ordinals are the ontological schema of the natural multiple.
An ordinal is a consistent natural unity, counted for one in the onto-
logical situation (set theory). These unities (in the non-numerical
sense of the pure and simple consistency of the multiple, of the 'gath-
ering together' of the multiples that constitute it, or belong to its
presentation) provide the material of Number, that on the basis of
which there is Number, or more precisely that within which Number
operates a section.
The simplest way to think about this is to con-
sider that a Number extracts a form from its natural ordinal material,
as a part, piece or fragment of it, a consistent unit of this material:
an ordinal.
12.8. Because of their antiquity, their universality, their simplicity
(which in fact masks a formidable complexity in the detail), the
natural whole numbers will be our guide. We have seen (chapter 11)
that, thought according to their being, natural whole numbers are
nothing but a particular section in the infinitely infinite domain of
ordinals: the section that retains only the initial point of being of this
domain (the void) and the 'first' successions, bounded externally by
OJ, the first limit ordinal. Or that the natural whole numbers extract
and isolate, in the boundless fabric of natural multiples, only that
which is finite.
Why not continue in the same way? It is certainly more rational
uniformly to attach the concept of Number to the ordinals in the
mode of a section, than to deploy an anarchical selection of disparate
procedures (algebraic, topological, set-theoretical ... , see 1.13).
Of course, we must be sure this is possible. 'Possible' meaning
what? That in this way we can find our familiar numbers. It would
certainly be arbitrary simply to impose, in the name of ontological
simplicity, a concept of Number which would not subsume either the
rational numbers or the real numbers. But if Number, as a section in
a natural multiple, defines whole numbers as well as rational (or
fractional) numbers, whole negative numbers as well as real numbers,
infinitesimals as well as ordinals, then nothing, in my view, can
prevail against both the mathematical unity and the philosophical
novelty of such a concept.
Moreover, the properly ontological simplicity of the idea of 'section'
confirms that our wager is good. To say that a Number is constituted
on the one hand by an ordinal (which is the signature of the Number's
belonging to the natural form of presentation), on the other by a part
of that ordinal (which is the section as 'formation' in the natural
material) is to define Number by putting to work only the most ele-
mentary, 'basic' categories of the ontology of the multiple.
12.9. Number will then appear as the mediation between Nature's
infinite prodigality of forms of being and that which we are in a
position to traverse and to measure. It is that which, at least in a
limited domain of its existence, accords our thought the capacity to
grasp and measure being qua natural being. Something which every
physics confirms.
12.10. There is no doubt that Aristotle's language (Matter and Form)
is the most eloquent one for transcribing the idea of Number. In
particular, it affords us the advantage of installing ourselves within
materialist metaphors. This is no negligible advantage when we know
that, since Plato, on account of its apparent mystery, Number has
been at the heart of all idealist representations of Nature. Up to, and
including, what it has become under the law of Capital, what it is
today, as I recounted at the beginning of this book: the unthought
basis of the ideology of the countable.
Since the section of Number always operates upon an ordinal, it
can be said that, given any Number whatsoever, there always exists
an ordinal that is its matter. 'Matter' here has a very precise meaning.
On the one hand, the ordinal is the 'basis' of Number, that from
which its form is sectioned. Thus it proceeds from one ordinal, from
which an extraction is made, that there should be a Number qua
principle of this extraction. On the other hand, we know that all the
elements of an ordinal are ordinals (see 8.5). If the numericality of
Number, what it sections, its form, is a part of an ordinal, then, since
all the elements of a part of a set are obviously elements of that set,
that which sections a Number must also be entirely composed of
ordinals. 'Matter' this time means first matter. When we speak of the
constituents of the numerical section, we are speaking exclusively of
ordinals. It is an ordinal that is sectioned, and the elements of the
section are also ordinals. With regard to the categories of its matter,
Number is natural through and through.
12.11. The Aristotelian metaphor is easily extended: we say that the
product of the numerical section, in the ordinal that indicates its
natural provenance and furnishes its matter, is the form of the Number.
Number itself is rather the gesture of sectioning, which is why it is
represented by the pair of its matter (an ordinal) and its form (a part
of that ordinal). But in the form is concentrated that by virtue of
which Number escapes its natural prescription, or at least might
escape it. Because the form, being any part of an ordinal whatsoever,
brings forth, within a natural unity, a multiple which in general is
not natural.
The form is, simply, a set of ordinals taken from among the ele-
ments of an ordinal. This deduction distinguishes a part of the matter.
Now, although every ordinal is a set of ordinals (in fact, the set of
ordinals which precedes it, 11.2), not every set of ordinals is neces-
sarily an ordinal. An ordinal has no holes; all ordinals that precede
it belong to it, from the void 0 right up to itself. This is, moreover,
why an ordinal is the name of its own 'length'. If, on the other hand,
you take any set whatsoever of ordinals, there is a good chance that
a great many ordinals will be missing, that the set will be full of holes.
It will therefore not itself be an ordinal. Consequently, the form of
a Number is usually not an ordinal; only its matter is. As might be
expected in a materialist philosophy, it is matter that is homogenous,
non-lacunary, regular, and form that is holey, irregular, non-natural.
With the form of a Number we generally transgress the limits of
natural being, even if its material is always extracted from within
those limits.
12.12. If the form is a part extracted from an ordinal by the section
which is Number - a (usually non-natural) subset of a natural
set - then it leaves a remainder; there is something like the leftover
cuttings from the sculpting of the form in the ordinal-matter. This
remainder is made up of those elements of the initial ordinal that are
not elements of the form of the Number, the portion of the matter
that is not taken up in the form. We call this the residue of the
Just like the form, the residue of a Number is a multiple made of
ordinals. And, again just like the form, it is usually not an ordinal (it
would be somewhat paradoxical if the residue was natural; it is so,
nevertheless, in the specific case where the form cuts all ordinals out
of the ordinal-matter without exception, starting from ordinal W).
The residue is obtained by the simple difference between the Matter
and the Form.
It might be objected that, in that case, form and residue are
interchangeable. And, in a certain sense, that is the case. Contem-
porary art has blindly thought this ambiguity in the composition
of Number, by exhibiting as new works the residue of works of
art whose form is outdated. What will ultimately discriminate
between the residue and the form of a given Number, though, will
relate to the law of order over Numbers, a law we shall study in
chapter 13.
Note once more that taking the form and residue together - the
union of the form and the residue - restores the matter, that is, the
ordinal we began with. The set-theoretical triplet of matter, form and
residue is all there is to the numerical section.
12.13. Armed with these remarks, we can outline our programme of
investigation into the concept of Number:
• Study what it is that makes the difference between two Numbers,
and understand the law of order that serialises them and without
which we would not, finite subjects that we are, have any hope
of progressing in our knowledge of them.
• Reconstitute algebra, the operational dimension of Number (addi-
tion, multiplication, etc.), without which, constrained above all
as we are by the ideology of the countable, no one would believe
that Number is a number. We will always hold firm to the point
that the being of Number precedes operations, that Number is
above all a thinking, on the basis of Nature, of a section that
extracts a form from a natural unity thinkable as the matter of
• Find again, in the infinitely infinite swarming of Numbers, in the
incredible prodigality of being in numerical form, our historical
numbers: natural whole numbers, relative whole numbers (nega-
tive numbers), rationals (fractional numbers), reals, ordinals ...
• Show that there exist infinitely more Numbers than we can know
or can handle, that our historical numericality is most impover-
ished compared to the excess of being in Numbers.
• Make sure, in this way, both that Number opens an authentic
space for thought and that this thought explains in terms of effec-
tive operations only a minute part of all the types of Numbers of
which multiple-being - as coupled to thought by set-theoretical
ontology - is capable.
12.14. This programme accomplished, we will taste the bitter joy of
Number, in both its thinkable and its unthinkable aspects. Number
will be entrusted to being, and we will be able to turn ourselves
toward the numberless effects of the event.
Additional Notes on Sets of Ordinals
Nt. The concept of Number makes central use of the concept of 'part
of an ordinal'; that is to say, of the concept of an arbitary set of
ordinals extracted from a given ordinal. Some remarks must be made
concerning the correct treatment of the notion 'set of ordinals', which
incorporates that of a 'part of an ordinal', since all the elements of
an ordinal are ordinals.
N2. For a set of ordinals to be an ordinal it is necessary and sufficient,
as we have noted, that it should have no holes, that no ordinal should
disrupt the chain of belonging that binds the ordinals to each other
up to the ordinal under consideration.
N3. Since belonging is a total order over the ordinals, every set of
ordinals is totally ordered by belonging. And this is the case whether
or not it has holes. If X is a set of ordinals and Xl and Xz are elements
of this set, then it is always the case either that Xl E Xz, or Xz E Xl,
or Xl = Xz. Thus the form and the residue of a Number are totally
ordered by belonging, just as its matter is. What makes the form and
the residue unnatural are the holes in them, not their order. The
universal intrication of natural presentation prescribes its law to all
the components of a Number. But what subtracts most Numbers
from the strictly natural domain of being resides in the lacks that
affect their form (and therefore their residue). A Number is non-
natural in so far as its natural fabric is perforated.
N4. Every set of ordinals has a minimal element. This results once
more from that important law of natural multiples, the principle of
minimality (see 8.10). Take X, a defined set of ordinals; and P, the
property 'belonging to X'. If there exists any ordinal that possesses
the property (it is sufficient for this that X should not be empty), then
there exists a smallest ordinal that possesses it. It is this smallest
ordinal that is the minimal element of X: it belongs to all the ordinals
of X, but no ordinal of X belongs to it.
The existence of a minimal element has nothing to do with whether
or not the set has any holes. Therefore one can always speak of the
minimal element of the form of a Number, or of the minimal element
of its residue. As to the minimal element of its matter, this is always
the empty set 0, since the matter is an ordinal.
N5. We must be very careful on the other hand to observe that a
given set of ordinals does not always have a maximal element. We
have already seen that a limit ordinal (which is a set of ordinals) has
no maximal element (9.14). A fortiori any set whatsoever of ordinals
may very well be infinitely 'open', containing no element that domi-
nates all the others.
N6. However, there always exists an upper bound of a set X of
ordinals. By 'upper bound' we understand the smallest ordinal to be
larger than every ordinal in X. Here again, the existence of an upper
bound is guaranteed by the principle of minimality. Let P be the
property 'being larger than all the ordinals that belong to X'. There
certainly exists an ordinal that possesses this property, unless X is
equivalent to the set of all the ordinals, which is inconsistent. There-
fore there exists a smallest ordinal which possesses the property P; it
is the smallest ordinal to be larger than all the elements of X, and
thus it is the upper bound of X. We will denote this upper bound by
N7. If a proper part of an ordinal (a part which is not the ordinal
itself, a truly partial part) is an ordinal, then it belongs to the initial
We have known for a long time that the converse principle is part
of the definition of ordinals. They are transitive, and so every ordinal
that belongs to an ordinal is also a part of it. We now want to show
that every ordinal which is a proper part of an ordinal belongs to it.
This comes back to saying that, between ordinals, the order of belong-
ing is equivalent to the order of inclusion.
Suppose an ordinal WI is a part of ordinal W
• Since
belonging is a total order over the ordinals, and since WI is different
from W
(it is a proper part of W
), there are two possibilities:
1 Either WI E W
, and the theorem is true, the ordinal WI which
is included in W 2 belongs to it, the part is also an element.
2 Or W
E WI' But, since WI is transitive, that would mean that
C WI' Now we know that WI C W
• If one set is included in
another, and the other included in it, then they are equal, as is
intuitively obvious, and as the reader can prove in one line. Now,
WI cannot be equal to W
, since it is a proper part of it. Thus the
first case must hold, and the theorem is proved.
So it is the same thing, when dealing with ordinals, to say that one
belongs to the other, and to say that one is included in the other. In
other words, if an ordinal represents (as a part) another ordinal, then
it also presents it (as an element). Which does not prevent an ordinal
from having some parts which are not elements. These parts will
simply not be ordinals either. This would be the case, for example,
with holey, lacunary sets, sets which begin in the middle of an ordinal
chain or only present separated elements, etc. In fact it is generally
the case with the form of a Number.
If, however, the form of a Number is an ordinal, then it follows
from the preceding arguments that not only is it a part of the matter
(the initial ordinal), but also an element of it. Then the form is of a
peculiar kind, like a 'corpuscle' of matter. In such cases, Number is
less a representation extracted from Nature than a simple natural
Difference and Order
of Numbers
13.1. A Number is entirely determined by its matter (the ordinal
from which its form is extracted) and its form. The residue is obtained
by taking the difference between the matter and the form. Because
of this, it is often convenient, as we have said, to write a Number in
the form N = (W,F(N)), where W is the ordinal-matter and F(N) the
form. The residue R(N), is equal to W - F(N).
Given these conditions, how can we think the difference between
two Numbers? It is natural to posit that they are identical if they
have the same matter and the same form. If they are not identical,
this could be:
• because they do not have the same matter. Take WI the ordinal-
matter of one, and Wz the ordinal-matter of the other. Two ordinals
like WI and Wz are ordered by belonging, but also, as we have seen
(N7), by inclusion: either WI C Wz, or W
C WI' Thus we can say
that in this case what differentiates WI from Wz is the set Wz - WI,
or WI - W
• Since all elements of an ordinal are ordinals, we can also
say that what differentiates WI from Wz - and thus the numbers NI
and N
, of which these ordinals are the matters - are the ordinals
which are elements of W
but not of WI (if WI C W
) or elements of
WI but not of W
(if Wz C Wd;
• because, having the same matter, they do not have the same form.
In this case, there are elements (and therefore ordinals, since the first
matter of a Number is composed of ordinals in its three components,
matter, form and residue) that are in the form of one but not in the
form of the other. But, since the matter is the same, every element of
the form of one which is not in the form of the other is in its residue:
if W E F(N d and W E F(N
), then W E R(N
). What differentiates
the two Numbers Nt and N
is the set of ordinals that are in the form
of one and in the residue of the other.
We can see then that the difference between two Numbers can be
understood in terms of ordinals. If an ordinal is in the matter of one
and not in that of the other, or if it is in the form of one and in
the residue of the other, it makes a difference between the two
13.2. Take any two Numbers whatsoever. We will say that an ordinal
w discriminates between these two Numbers if it is in the matter of
one and not in that of the other, or if it is in the form of one and in
the residue of the other (which implies that it is in the matter of both,
since form and residue are both parts of matter).
13.3. Let's take an example: Let Nt be the Number (2,(0)) whose
ordinal-matter is 2 and whose form is (0). It is certainly a Number,
since 2 is an ordinal (it is the finite ordinal whose being is (0,(0)), see
11.5) and the singleton of 0, denoted by (0), is a part of that ordinal
(7.11). This Number Nt has, for matter, the ordinal 2, and, for form,
the part (0).
Now let N2 be the Number (0), 2). Once again it is a Number,
since 0) is an ordinal (the first limit ordinal) and the ordinal 2,
which is an element of 0), is also a part of it (transitivity of ordinals).
This Number N2 has for its matter 0) and for its form the part
(0,(0)) = 2.
The ordinal 0) does not discriminate between the Numbers Nt and
• Indeed, 0) is certainly not in the matter of Nt (which is 2, a finite
successor ordinal), but neither is it in the matter of N
, because this
matter is 0), and we know that no set belongs to itself: it cannot be
that 0) E 0).
The ordinal 0 (the empty set) does not discriminate between the
Numbers Nt and N
either. In fact, it is in the form of both. The
form of Nt is the singleton (0), of which 0 is the only element. So 0
is an element of this form. And, on the other hand, 0 is an element
of the ordinal 2, which is the form of N
• Thus 0 is also in the form
of N
However, the ordinal (0) (which is the whole number 1) does
discriminate between the Numbers Nt and N
: (0) is an element of
the ordinal 2, and thus belongs to the form of N
• But it cannot
belong to the form of N" which is precisely (0), since the self-belong-
ing (0) E (0) is impossible. Given that (0) is an element of the matter
of N, (which is the ordinal 2), since it is not in its form, it must be
in its residue.
13.4. Given two Numbers and any ordinal whatsoever, it is always
possible to say whether this ordinal discriminates between the two
Numbers or not. If N, and N
are Numbers, the property 'discrimi-
nating between N, and N
' is well-defined.
But if there is an ordinal that discriminates between N, and N
(that is, if N, and N2 are different), then in virtue of the principle of
minimality - which we have constantly made use of because it is a
fundamental law of natural multiples (see 8.10) - there is one unique
smallest ordinal which discriminates between them. Or, if you like,
a minimal ordinal for the property 'discriminating between the
Numbers N, and N
13.5. An extremely important definition: The smallest ordinal to
discriminate between two Numbers is called their discriminant.
The interesting thing about the concept of discriminant is the fol-
lowing: it brings the idea of the difference between two Numbers
down to a matter of one single ordinal. This 'minimal point' of
differentiation allows a local rather than global treatment of the
comparison between two Numbers. The existence of a discriminant
suffices for us to conclude that two Numbers are different.
13.6. One more example. Take the two Numbers N, and N
the example above (13.3), N, = (2,(0)) and N
= (00,2). What is their
• We have seen that 0 does not discriminate between N, and N
• We have also seen that (0) discriminates between them. Since the
only ordinal smaller than (0) is 0, which does not discriminate
between N, and N
, (0) is definitely the smallest ordinal that
discriminates between them. We therefore say that (0) is the dis-
criminant of the Numbers (2,(0)) and (00,2).
Note the location of the discriminant: it is in the matter of the
two Numbers N, and N
, but is in the form of N
and in the residue
Now consider the following two Numbers (S(W) denotes the
successor of the ordinal W, see 9.5):
• N3 = (S(ro),ro). Its matter is the ordinal S(ro), its form ro itself. The
latter is a part of S(ro), since every ordinal is an element of
its successor, and every element of an ordinal is a part of it
(transitivity) .
• N4 = (S(S(ro)),ro). Its matter is the successor of the successor of ro,
its form is roo
What is the discriminant of N3 and N
? These two Numbers have
the same form, that is ro, but extracted from different matters, S(ro)
and S(S(ro)). In summary, in these numhers everything is exactly the
same up to the ordinal S(ro). This ordinal is in the matter of N
, since
S(ro) E S(S(ro)), but it is not in the matter of N
, since S(ro) fl: S(ro).
Thus the ordinal S(ro) is the smallest ordinal to make a difference
between N3 and N
; it is the discriminant of these two Numbers.
Note once again the location of the discriminant of N.J and N
S(ro) is not in the matter of N
, but is in that of N
• Meanwhile,
this time it is not in the form of N
, which is roo It is therefore in
its residue.
The comhination between the ordinal punctuality of the
discriminant and its location in the Numbers compared will give
us the key to the concept of order in the boundless domain of
13.7. Let's give all of this a slightly stricter form.
The location of an ordinal w with regard to Number N, written
L(w,N), is the position that it occupies with regard to the three
dimensions of the numerical section carried out by the Number N:
matter, form, residue. There are obviously three locations:
1 Either the ordinal w is not an element of the ordinal W which is
the matter of the Number N. In this case we say that it is located
'outside the matter' and we posit that: L(w,N) = oM(N).
2 Or the ordinal w is in the matter Wand belongs to the form of
the Number. We then posit L(w,N) = F(N).
3 Or the ordinal w is in the matter W, but belongs to the residue
of the Number: We then posit L(w,N) = R(N).
When there is no ambiguity as to the number N in question, we
might simply use the notation L(w) = R, signifying that the location
of w (for the number in question, of course) is its belonging to the
residue (of that number).
Given a number N, every ordinal can be located for N so long as
we allow the location 'outside the matter'.
When an ordinal discriminates between two numbers Nt and N2
(see 13.2), it is very simply because its location in the two Numbers
is not the same. The table of possible locations for an ordinal W which
discriminates between the two Numbers is as follows (using oM, F
and R to denote the locations):
L (w,Ntl L (w,N
F oM
R oM
oM F
oM R
The discriminant of Nt and N
, being the smallest ordinal to dis-
criminate between them, necessarily responds to one of the 'pairs' of
locations indicated in the table. For example, if it is in the residue of
Nt. it must be in the form of N
or outside the matter of N
, etc.
13.8. Definition of order over Numbers
Take two Numbers Nt and N2 and their discriminant w (if neces-
sary, reread 13.4-13.6, given that the concept of discriminant is
We say that Nt is smaller than N
, written Nt < N
, if the location
of the discriminant w for the Numbers Nt and N2 satisfies one of the
three following cases:
1 Either L(w,Nd = R(Nd, and L(w,N
): the discriminant
is in the residue of N
and in the form of N
2 Or L(w,N tl = oM(N tl, and L(w,N
): the discriminant is
outside the matter of NJ and in the form of N
3 Or L (w,Ntl = R(Ntl and L(w,N
) = oM(N
): the discriminant is
in the residue of NJ and outside the matter of N
Compare these three cases carefully with what the table in 13.7
indicates as to the possible locations of the discriminant of two
13.9. It is not immediately evident that the relation Nt < N2 is one
of order. But, even before establishing that this is the case, we can
reveal the characteristics of this relation.
The discriminant gathers into one point (one ordinal) the concept
of difference between two Numbers. The order introduced here
depends on the location of this point, and therefore on a sort of
topology of difference. Since, in the gesture of sectioning that consti-
tutes every Number N, the 'positive' numericality - that which this
gesture extracts from matter - is the form, we will always consider
that, if the discriminant of two Numbers is in the form of one, this
number is 'larger' than the other. In the other, of course, the discrimi-
nant will either be in the residue or outside the matter.
Conversely, the residue of a Number is the purely passive result of
the section of its form, the unintentional remainder of the numeric
gesture. It is that which Number as gesture leaves to matter. If the
discriminant of two Numbers is in the residue of one of them, we
will always consider this Number to be 'smaller' than the other.
In the other, the discriminant will be in the form, or outside the
13.10. An apparently paradoxical consequence of this conception,
which determines all order on the basis of the active superiority
of form - thought as the numericality of Number - over residue -
thought as passive inverse - is that a number NI is said to be smaller
than N
if their discriminant is in the residue of N I and outside the
matter of N
• The 'paradox' results from the assumption that the
position 'outside matter' is completely unaffected by the numerical
gesture, being neither in its form nor in its residue. Isn't it even more
passive then, even less involved in the numerical extraction of the
form, than an ordinal which is in the residue, and which therefore at
least figures in the matter of the Number? Isn't the location oM a
figure of nothing in relation to Numher, an ontologically 'inferior'
figure to the passive figure of the residue?
13.11. This sense of 'paradox' misses an essential point, which is
that the 'outside matter' position includes the matter itself, since an
ordinal W is not an element of itself. There is no reason to suppose
that the matter is 'indifferent' to the gesture of Number: it is its
primary 'given', that on the basis of which there is Number; the
natural multiple whose being is exposed to the numerical section.
And it is always the matter itself of one of the two Numbers that is
at stake when the discriminant is located 'oM' for one of them.
If the discriminant of NI and N
is, say, outside the matter of N
then it is in the matter of N I (in its form, or in its residue). If not - if
it were outside the matter of both Numbers - it could not discrimi-
nate between them. Therefore the discriminant must indeed be the
smallest ordinal in the matter of N, and outside the matter of N
Evidently this means that the ordinal WI which is the matter of N,
is larger than the ordinal W
which is the matter of N
• If not, there
could be no ordinal in W, that was not in Wz, since the elements of
an ordinal are all the ordinals that precede it (see 11.2). This means
that W
E W, (the order-relation over the ordinals is belonging). But
Wz itself is the smallest ordinal that does not belong to Wz, since all
the ordinals smaller than Wz are, precisely, the elements of Wz. And
so, ultimately, Wz is in W, and is the smallest ordinal not to be in
Wz. It is the smallest ordinal to be in W, (the matter of Ntl, and not
in Wz (the matter of N
), and therefore outside the matter of N
. Wz,
the matter of Nz, is the discriminant of NI and N
This demonstration has a general validity: whenever we say that
N, is 'smaller' than N
, or that N, < N
, because the discriminant of
N, and N2 is in the residue of N, and outside the matter of N
, this
also means that the discriminant in question is the ordinal-matter of
• And this relation is legitimate because the matter of a Number,
the one-ordinal in which the numerical section operates, is a primor-
dial donation of being ontologically superior to the passivity of the
It is therefore philosophically well-founded to put the locations
in the order R < oM < F: the form, affirmative numericality of the
section, is superior to what is outside matter, which is itself superior
to the passivity of the residue because in reality this 'outside of
matter' is the matter itself, integrally counted for one as an ordinal.
The relation N, < N2 founded on otlr three cases (the discriminant
in R(Ntl and F(N
); the discriminant in R(Nd and oM(N
); the dis-
criminant in oM(Nd and in F(N
)) describes a hierarchy, founded in
the being of Number as the sectioning of a form in natural matter.
13.12. What remains now is to establish that the relation N, < N
truly an order-relation, in the mathematical sense of the term: that it
serialises Numbers. This amounts to responding positively to three
1 Is the relation total?' Or: given any two different Numbers N,
and N
, is it always the case that N, < N
or N2 < Nt?
2 Is the relation non-reflexive? Or: is it impossible that Nt < Nt?
3 Is the relation transitive? Or: from the relations Nt < N2 and
< N3 does it necessarily follow that N, < N3?
If we prove these three points, we will have brought the
philosophical legitimacy established in 13.11 to coincide with
mathematical (ontological) legitimacy. Or, rather, with regard to the
order of Numbers, we will have obtained the situation in which we
have been continuously striving to remain: where what is said under
the sign of the philosophical statement 'mathematics is ontology'
remains in harmony with what is said under the sign of mathematical
inferences themselves. Or where the interpretation of mathematics as
science of being qua being draws its contact with the real from the
effective thoughts of such a science.
13.13. The relation < is total
Look one last time at the table of cases of inequality for N 1 < N
This table fixes the location of the discriminant of Nl and N
Case 1
Case 2
Case 3
To demonstrate that the relation is total is to show that, given two
different Numbers, one of them is always 'smaller' than the other.
Take two randomly selected numbers N3 and N
, and w, the
ordinal which is their discriminant. Three cases are possible:
1 The discriminant w is in the residue of N
. Then:
(a) either it is in the form of N
, and (see the table) N3 < N
(b) or it is outside the matter of N
, and (idem) N3 < N
2 The discriminant w is in the form of N
. Then:
(a) either it is in the residue of N
, and (idem) N4 < N3;
(b) or it is outside the matter of N
, and (idem) N4 < N
3 The discriminant is outside the matter of N
. Then:
(a) either it is in the form of N
, and (idem) N3 < N
(b) or it is in the residue of N
, and (idem) N4 < N
Having exhaustively enumerated all the possible cases, we see that
the relation < between Numbers N3 and N4 is always defined. The
relation really is total in the domain of Numbers, there are no two
different Numbers not related by <.
13.14. It is good to get into the habit of thinking through the inequal-
ities between Numbers more rapidly. For example we could say: if
the discriminant w is in the residue of one of the two Numbers, the
table (cases 1 and 3, which are the only possibilities) shows that this
Number is smaller than the other. If w is in the form of one of the
two Numbers, the table (cases 1 and 2, the only possibilities) shows
that this number is larger than the other. Apparently we have left to
one side the case where w is outside the matter of one of the Numbers.
Not so, because then it would necessarily be in the residue or the
form of the other (if it was outside the matter of both, it would not
discriminate between them), and we are referred back to one of the
preceding cases.
To compare two numbers according to the relation <, we therefore
proceed as follows: firstly we check whether the discriminant is in
the form of one of them: if so, we conclude immediately that this
Number is the largest. If not, we check whether it is in the residue
of one of the two: if so, we conclude that that Number is the small-
est. The work is done, no other case is possible.
13.15. The relation < is irreflexive
This point is trivial. It cannot be that N 1 < Nt. since the relation is
founded on the existence and location of a discriminant, which cannot
exist 'between' N, and itself.
13.16. To exercise ourselves in the comparison of Numbers using the
< relation, let's take up the examples from 13.6 once more. We had,
adopting the notation by the pair of matter and form, the four
following Numbers:
N3 = (S(oo),oo)
N4 = (S(S(oo)),oo)
The discriminant of N, and N2 is (0). It is in the residue of N"
and in the form of N
. So N, < N
The discriminant of N3 and N4 is S(oo). It is outside the matter of
, and in the residue of N
. So N4 < N
The discriminant of N3 and N, is (0), which is in the residue of N,
and in the form of N
. So N, < N
The discriminant of N4 and N2 is 2 (why?). It is in the residue of
N2 and in the form of N
. So N2 < N
The reader can study the remaining comparisons on their own
It will be remarked that it is not simply because the matter of a
Number is 'larger' that that Number is larger. Thus N4 has for
ordinal-matter the successor of the successor of 00, which is larger
than the successor of 00, the matter of N
. Nevertheless, N4 < N ~
What is still more remarkable in this example is that the form of
N3 and of N4 is the same (it is (0). Thus we have the following 'law':
if the form stays the same whilst the matter grows, the Number gets
smaller. It is quite straightforward to demonstrate the general
case. Take a Number NI (W"X) and a Number N
(Wz,X), where
; and X the same set of ordinals (which is a common part
of WI and W
). The discriminant of these two Numbers cannot be
found in the form of either of them, since they have the same form,
X: an element of X cannot discriminate between them (it even has a
location for N I and N
, namely the form). It is therefore in the residue
of one, and outside the matter of the other. It is clear that this
discriminant is none other than WI, which is the smallest ordinal not
to belong to WI, and which is in W
, since WI E W
• Now WI is
necessarily in the residue of N
(since it belongs to its matter, but not
to X, its form), and outside the matter of N
. Therefore it is indeed
the case that N
This process suggests a comparison between the Number (W,X)
and the relation ~ We know that such a relation diminishes when
its denominator W grows. But be warned: this is only a distant
analogy, because ~ means nothing here.
All the same we can show, inspired by this analogy, that, if the
matter remains the same whilst the form gets larger - so that the old
form X is included in the new form x' - then the Number gets larger.
This time, it is the enlargement of the 'numerator' that enlarges the
'relation'. I leave the details of the demonstration of this to the reader.
Suffice to say that the discriminant is the smallest ordinal to belong
to X' and not to X; so it is in the form of the second Number and
in the residue of the first, and therefore the second is larger.
These observations are philosophically well-founded. What does
it mean, in fact, to produce the same form from a larger matter? That
the gesture of the numerical section did not manage to extract from
a vast matter (that of a larger ordinal) any more of a form than could
have been obtained with a smaller matter. The gesture was thus less
concentrated, less elegant, less effective. It is quite legitimate that the
Number which marks this gesture should be held for inferior. The
converse also follows: to obtain a more widely deployed form, con-
taining all the elements of the first and more, with the same initial
matter, requires a more efficient gesture of sectioning. It is quite right
that this should be marked by a superior Number.
The relation < does indeed express in the mathematical field the
ontologically rational dispositions of the comparison of Numbers.
13.17. The relation < is transitive
This is a question of proving that, given three Numbers, Nb N2 and
, if NJ < N2 and N2 < N
, then NJ < N
. Obviously everything
hinges on the location of the discriminants. We shall write the
discriminant of N
and N2 as w(1,2), that of N2 and NJ as w(2,3),
and that of NJ and N
as w(1,3).
(a) First step. An ordinal smaller than w(1,2) and w(2,3) does not
discriminate between N
and N
The discriminant of two Numbers is the smallest ordinal that dis-
criminates between those two Numbers (in the order of ordinals,
which is belonging).
If an ordinal W is smaller than w(1,2), it doesn't discriminate
between Nl and N
. Its location (F,R, or oM) is the same in N
in N
. Equally, if it is smaller than w(2,3), it doesn't discriminate
between N2 or NJ either - its location is the same in N2 and N
Ultimately, therefore, its location must be the same in Nb in N2 and
in N
, and it does not discriminate between NJ and N
(b) Conclusion of the first step: w(t,3), which obviously discri-
minates between Nl and N
, cannot be smaller than w(1,2)
and w(2,3). It is therefore at least equal to the smaller of the two.
(c) Second step. The smallest of the two ordinals w(1,2) and w(2,3)
discriminates between NJ and N
For convenience of exposition, we will suppose that the smallest
is w(1,2) (the reasoning would be exactly the same if it was w(2,3);
confirming this would be an excellent exercise for the reader). Since
w(1,2) discriminates between NJ and N
, its location in NJ is different
from its location in N
. But, since it is smaller than w(2,3), it does
not discriminate between N2 and N
, since w(2,3) - discriminant of
N2 and N3 - is the smallest ordinal that discriminates between these
two Numbers. Therefore the location of w(1,2) in N2 and N
is the
same. Since its location in N2 differs from that in N b if it is the same
in N3 as in N2, its location in NJ also differs from its location in N
So w(1,2) discriminates between Nl and N
(d) Third step. w(1,3), the discriminant of Nl and N
, is actually
equal to the smallest of the ordinals w(1,2) and w(2,3).
We have seen that w(1,3) must be at least equal to the smallest of
the two ordinals w(1,2) and w(2,3) (first step). We supposed w(1,2)
to be the smallest. Thus w(t,3) is at least equal to w(1,2). Now w(1 ,2)
discriminates between Nl and N3 (second step). Since w(1,3) is the
discriminant of N, and N
, and thus the smallest ordinal to discrimi-
nate between them, and since it cannot be smaller than w(l,2), which
discriminates between N, and N
, it is equal to w(1,2). So w(l,3) =
(e) An aside: if we were to suppose the opposite hypothesis, that
w(2,3) is smaller, we would find that w(l,3) = w(2,3), for the same
(f) Fourth step, the conclusive step. We have discovered that w(l,3)
= w(l,2). This can be expressed as follows: w(1,2), discriminant of
N, and N
, is also the discriminant of N, and N
Now, we know that N, < N
. So we know there are two possible
locations for the discriminant w( l,2) in N" the smaller of the two
1 Either w(l,2) is in the residue of Nt. But then, since it is also the
discriminant of Nt and N
, its position in the residue of Nt leads
us to conclude that Nt < N3 (on this point, see 13.14).
2 Or w(l,2) is outside the matter of Nt. It must then be in the form
of N
, for the usual reason that Nt < N
. But w(l,2), which is
smaller than w(2,3), does not discriminate between N2 and N
Therefore it is also in the form of N
. And, since it is the discrimi-
nant of N, and N
, being outside the matter of Nt and in the form
of N
, once again Nt < N
So we have proved that, if Nt < N2 and N2 < N
, then N, < N
We have even discovered, as a bonus, a still finer result: the discrimi-
nant of Nt and N3 is equal to the smallest of the discriminants of Nt
and N
, and of N2 and N
13.18. Dialogue with a tenacious reader, on the subject of the preced-
ing demonstration:
THE READER: You suppose from start to finish that the discriminant
of N, and N3 exists. It's not so obvious. I can well see that the dis-
criminant of Nt and N2 exists, since we know that N, < N
. The same
for that of N2 and N
. But it could well be that in the end N, = NJ,
and in that case there would be no discriminant w(l,3). The relation
would be circular: Nt < N2 < N,.
ME: But that's absurd: If the discriminant of N, and N2 is located in
N, and N2 in such a way that N, < N
, it cannot be the case that
N2 < N,. Therefore N, is necessarily different from N
, and their
discriminant exists.
READER: Okay, you've got me. But I'm still not satisfied. In your
second step, you suppose that one of the two discriminants
w(1,2) and w(2,3) is the smaller of the two. But surely it could quite
easily be the case that there is no smallest of the two; for this to be
the case it suffices that they be equal. And, so that you don't try to
pull the wool over my eyes, I'll give an example. Take these three
1 N, is the number (2,(1)), which has for its matter 2 and for its
form the singleton of 1. I know (I've read you saying so just now)
that the ordinal 1 is an element of the ordinal 2 (see 11.5), and
that the singleton of an element is a part (7.10). Here we have
the pair of an ordinal and of a part of that ordinal, so it's a
Number (12.1).
2 N2 is the Number (0,0), which has for matter the empty set, and
for form the empty set. It's still a Number though! Because 0 is
an ordinal, which serves for the matter, and 0 is a universal part
of every set (see 7.9), including 0 itself, which is, as I know, a set.
So 0 is fine as the form too.
3 N3 is the Number (2,1). You can't refuse me this, because 2 is an
ordinal, and 1, being an element of the ordinal 2 (following your
11.5, as always), is also a part of it, since every ordinal is
Now, let's see, what do I have? The discriminant of N, and N2 is
0: it's in the residue of N" since 0 is an element of the matter 2, but
does not figure in the form, the singleton (1), whose only element is
1. And it's outside the matter of N
, since this matter is 0, of which
o cannot be an element. Therefore N, < N
The discriminant of N2 and N3 is also 0, which is outside the
matter of N
, as we can see, and which is in the form of N
, since
o E 1. From this we conclude that N2 < N
Here is a concrete test case where N, < N
, where N2 < N
, and
where, nevertheless, w(1,2) and w(2,3), to use your notation from
the beginning of 13.17, are equal. Therefore neither is smaller than
the other, and your chain of inference is broken.
ME: Very shrewd! You will have to allow me all the same that in the
end, in your example, transitivity is confirmed. Because the discrimi-
nant of N, and N3 is still 0, which is in the residue of N, and in the
form of N
. So it is still the case that N, < N
READER: I make an objection on a point of principle, and you respond
with an empirical remark! My example ruins your general argument,
which rests on the fact that one can always discern the smallest of
the discriminants of N, and N
and of N2 and N
. I have shown you
a case where this cannot be done. The fact that transitivity still works
for my example might just be chance, since it now seems you have
yet to prove anything.
ME: You allow my first step, all the same: that w( 1,3) cannot be
smaller than w(1,2) and w(2,3)?
READER: With the caveat that the 'and' seems somewhat suspect to
me, since it might relate two equal discriminants. See my example:
you would be saying that 'w(1,3) cannot be smaller than 0 and 0',
which is ludicrous.
ME: Unless it was smaller than 0 ... But anyhow - if, as in your
example, w(1,2) is equal to w(2,3), do you admit that w(1,3) cannot
be smaller than w(1,2) alias w(2,3)? Because no ordinal smaller than
this common discriminant can discriminate between N" N2 and N
READER: Obviously.
ME: But w(1,2) discriminates between N, and N2 - its location in N,
isn't the same as in N2?
READER: No; how could it be?
ME: And it also discriminates (going by the name of w(2,3), to which
it is equal) between N2 and N3 - its location is not the same in these
READER: That's exactly what I said.
ME: Let's look at these locations more closely. Since N, < N
, w(1,2)
must either be in the residue or outside the matter of N,. But can it
be outside the matter?
READER: (after some time thinking) No. Because, if it were outside
the matter of N" it would have to be in the form of N
, since N, <
• But, as it is also the discriminant of N
and N
, and N2 < N
, it
cannot be in the form of N
, as explained in 13.14. So it is definitely
in the residue of N, and ...
ME: ••• outside the matter of N
, because not in its form. But where
is it located in N3?
READER: (after some time thinking) In the form. Because this w(1,2),
which is also w(2,3), is the discriminant of N2 and N
. Being outside
the matter of N
, since N2 < N
, it is in the form of N
ME: Perfect! Here is a w(1,2) which w(1,3) cannot be less than, and
which is found in the residue of Nt and in the form of N
. Therefore
it discriminates between Nt and N
. That is to say ...
READER: Okay, I get it. Already identical to w(2,3), it must also be
identical to w(1,3). And this identity means that Nt < N
, since this
common discriminant (of NI and N
, of N2 and N
, and of Nt and
) is in the residue of Nt and in the form of N
• That works.
ME: It's just as your example says: 0 was the common discriminant
of the couples N t-N2 and N
. It is also the discriminant of Nt
and N
. And it is located in the residue of N h outside the matter
of N2, and in the form of N
. Which gives us the sequence: Nt < N2
READER: You must admit that you've had to add quite a bit to your
original account.
ME: It is a subsection of the argument, the principle is the same. But
in mathematics one cannot skip over anything, for the reason that
one never knows what one is skipping over.
13.19. Since the relation < is total, irreflexive and transitive, it really
is an order-relation in the mathematical sense. We have entirely justi-
fied our saying 'Nt is smaller than N
' when it is confirmed, by means
of the location of the discriminant of N I and N
, that the relation
Nt < N2 is valid.
Thus the universe of Numbers - even if it is, as we shall see,
borderless, saturated to an inexpressihle degree, of a density with
regard to which the celebrated 'continuum' is thin and lacunary - can
nevertheless be comprehended wholly under the serial law of an
The additional fact that this order can be designated solely by the
examination of the location of an ordinal (the discriminant) with
regard to three possible sites (F, R and oM) indicates a simplicity that
is reassuring as to our capacity to think the universe of Numbers.
It is striking that, given its combination of a logic of minimality
(the discriminant, the smallest ordinal to mark the difference of two
Numbers) and a logic of positions (the three components of the
numerical section), this order appears to be allied with lexicographi-
cal order. In fact, it is presented as such in purely mathematical
Now, lexicographical order, which organises words by recourse to
an alphabet of the phonic or scriptural unities that compose them,
touches on the distinction, so important in Lacan, between the signi-
fier and the letter.
In reality, Number is indeed like a signifier, whose
internal 'positions' are the three locations - matter, form and residue
- and whose letters are the ordinals. This alone permits us to organise
something as anarchic as sets of any ordinals whatsoever, ordinal
If Number is the medium in which Nature, grasped in its being,
opens itself to our thought, this is, without doubt - as the order of
Numbers testifies - because, in the section it carries out, we find,
under the simple form of one and three, that dialectic of the position
and of the letter which has been recognised, since Galileo, as the true
terrain of materialism. Nature consents to its profusion within the
fiction of a writing system; and we must recognise in Number the
most inscribed instance of being:
two fingers
snap in the abyss, in
a world rushes up, this depends
on you.
The Concept of Sub-Number
14.1. The concept of substructure, and even (in category theory) I
that of the sub-object, is fundamental for all areas of contemporary
mathematics. We know the extreme importance of the determination
of subgroups of a group, subs paces of a topological space, etc. A
good many of the most profound mathematical theorems of recent
years are theorems of decomposition or of presentation: proving that
a structure can be presented as a composition of (possibly simpler)
substructures, or that a structure is decomposable into a sequence of,
or as a product of, pre-defined substructures. The elegance of thought
reaches its highest point when one manages completely to 'resolve' a
presented axiomatic structure into substructures that are of the same
type, but simpler. Finite group theory offers some spectacularly
accomplished examples of such resolution.
The underlying idea is as follows: since the 'material' of mathemat-
ics is the pure or undifferentiated multiple, structures are inevitably
homogenous with structured sets. Mathematical ontology is unitary:
there aren't, on the one hand, pre-given 'objects', on the other, struc-
tural relations into which these objects enter. Everything can poten-
tially be reduced to a multiple without quality, made of the void
alone. Given this fact, it is inevitable that the exercise of thought
should consist in reducing complex multiplicities to simpler multi-
plicities, through the medium of the axiomatic definition of simple
and complex. The concept of structure organises this medium: it dis-
tinguishes elementary configurations from more intricate configura-
tions. Ultimately the strategic stakes of the thinking of being qua
being are to discern - given that every multiple is a multiple of mul-
tiples (One having no being) - which multiples a presented multiple
assures, in its turn, the presentation of. Whence theorems of decom-
position, resolution, or presentation.
What a mathematician calls an 'object' is nothing but a multiplicity
within which sub-multiplicities are intricated, often in a very opaque
fashion. The object is a packet of multiples, whose intrication is an
obstacle to thought, and within which must be separated, as far as
possible, the multiples-regions whose presentational combination is
assured by the 'object'. The 'objective' illusion, what we might call
the phantasm of the object, relates to the initial distance between the
entanglement of multiples and our separative access to this entangle-
ment through the medium of language. Concepts, axiomatically
introduced, determine types of structures, which are the operators of
separation and allow us to exhibit such an 'object' as an articulation
of substructures, indicating the latency of sub-multiplicities in their
relation to the medium of language.
That a structure can be resolved into substructures according to
various operators of combination (embedded sequences of subgroups,
finite or infinite products of compact spaces, etc.) is the definitive
mark, in the inscribed strategy of thought, of the fact that what it
confronts is being qua being, in the figure of an infinite entanglement
of pure multiplicities. A mathematician will say that he has 'thought
the object' (or 'understood the problem') when he has mapped the
linked immanence of the substructures whose presentative bond is
detained, initially in an opaque fashion, by the 'object'. So it is also
a question of the decomposition of the object, a putting to death of
the phantasm of the object, which is an object only in so far as it
resists, through its constitutive entanglement, its resolution into the
specific diversity of structures. Thinking hy means of substructures
deposes the object and returns toward being.
14.2. In its commonly accepted usage, the concept of number is not
a concept of the structural type. One doesn't speak of 'numerical
structure' as one speaks of the structure of groups or of vector space.
What is called 'number-theory' today is an inconsistent set whose
centre of gravity is in fact a certain area of algebra: ring theory and
ideals theory. In particular, no concept of sub-number exists, since
'number' doesn't designate a type of structure.
Consequently, since the Greeks, the concept of number has been
the principal redoubt of a realist, even empiricist, vision of mathemat-
ics. Either number is taken for a 'given' entity, or taken as proof that
mathematical nominations have a strictly symbolic or operational
value. There is a c1osing-in-on-itself of the entity 'number', which
is linked to its purely algebraic manipulation. Certainly, numbers
are combined according to algebraic rules. But it does not at
all follow from 7 + 5 = 12 (whether this statement be analytic or
synthetic) that 7 and 5 are 'substructures' of 12. The most tenacious
illusion of objectivism resides in the conviction that 7, 5 and 12 are
non-decomposable marks, whose serial engenderment assures their
It would therefore be a great victory for an ontological vision of
mathematics to establish the structural character of number, to
unbind it from its empirical punctuality, to extract it from the simple
form of the object. This programme, which would make of the predi-
cate 'number' a reputable type of pure multiple, would find its most
significant moment in the determination of the concept of sub-number.
This concept would align numericality with the great structural cat-
egories of mathematical thought (group, field, space ... ); categories
by means of which thought separates and unbinds the intrications
of the pure multiple.
14.3. The set-theoretical presentation of the concept of Number,
such as we have worked it through above, authorises a strict defini-
tion of the sub-Numbers of a given Number. Better still: as we shall
show step by step, a Number is defined in a univocal manner by
its sub-Numbers. There exists a presentation of Number on the basis
of the Numbers that are immanent to it. Thus Number in its turn
admits of theorems of decomposition or of presentation. It is
14.4. The concept of sub-Number
The general idea of the sub-Number is very simple: we obtain a sub-
Number of a given Number if we 'partition,2 this Number at a point
of its matter and keep everything that comes 'before' this partition.
Since the matter of a Number is an ordinal, a 'point' of partition is
an element of this ordinal, and thus a smaller ordinal. What there is
'before the partition' is constituted by the ordinals smaller than the
one that defines the partition. But the ordinals smaller than a given
ordinal are precisely the elements of that ordinal. Consequently, if w
is the point of the partition, then what comes before it, being consti-
tuted by all the elements of w, is nothing other than w itself. By
partitioning at point w we obtain a new ordinal-matter, which is w
- a matter evidently more limited than that from which it was cut.
But, it will be asked, what happens to the form, the numerical
section from the matter? Here, once again, the idea is very simple: as
the form of the new Number, we keep precisely those ordinals that
are in the form of the partitioned Number and which are 'before' the
partition. A sub-Number will truly be a segment of a Number, up to
point w, retaining up to w (that is, between 0 and w) all the charac-
teristics of the partitioned Number.
Let's give all of this a more precise form. Take a Number
N, = (W),F(Nd) and an ordinal w which is an element of WI (i.e. is
in the matter of Nd. We partition NI at point w, retaining only
ordinals that are lower than w, without changing the rest at all: the
elements of the form of the new Number will be those of F(Nd that
are lower than w. We thus make use of a property possessed by every
set of ordinals (and therefore by the form of every Number) (see N3):
because it is composed of ordinals, its elements are ordered by the
relation E • It is therefore entirely proper to speak of 'all the ordinals
of F(Nd smaller than the ordinal w'. The diagram (compare 12.4)
shows this:
Number: ••
Sub-Number: •• -------------.:----------------
We write Elw for the segment up to point w of a set E of ordinals
of which w is an element. Elw contains only elements of E lower than
w (but not w itself, please note!). The Number obtained by the parti-
tion of N I, and which, by extension of our notation, we will call N Ilw
(which means that w must be in the matter W, of N), that w E Wd,
will have as its code: (w,F(N dlw). Its matter is w - the point at which
it is partitioned, an ordinal that comes 'before' WI - and its form is
composed of all the ordinals in the form of N I which are smaller than
w. By the same token, its residue is composed of all the ordinals
smaller than w which are in the residue of N
We should note that this Number (w,F(Ndlw) is exactly 'like' NI
up to the ordinal w (exclusive): in fact, up to w, any ordinal that is
in the form of NI is in the form of (w,F(Ndlw) too, and an ordinal
that is in the residue of the former is also in the residue of the latter.
The new Number obtained through partition is, in short, the 'initial
segment' of N), an exact copy of the 'beginning' of N
Take two Numbers N, and N
. If there exists an ordinal w such
that NI = Niw, where NI partitions N2 at point w, then we say that
N, is a sub-Number of N
• Or, alternatively: a sub-Number of N, is
a segment N "w of N ,.
14.5. One sub-Number of N, - and one only - can be defined for
every ordinal w in the matter of N,: therefore for every element of
WI' There exist exactly W, sub-Numbers of N" since an ordinal
'counts' the ordinals that precede it. Generally speaking, a Number
admits of as many sub-Numbers as there are ordinals in its matter.
14.6. Take N,lw, a sub-Number of N,.1t is clear (see the definitions
and the diagram) that w is the discriminant of N"w and N" since,
up to w, these two Numbers are identical. Now, the matter of N"w
is w. So w is outside the matter of N,lw. The order-relation between
N, and its sub-Number N,lw will therefore depend entirely upon the
location of the ordinal w in the Number N,: whether w is in its form
or in its residue.
There are therefore two types of sub-Numbers for a given Number
1 Sub-Numbers N,lw2 where W2 - which is at once their matter and
the discriminant of themselves and N, -is in the form ofN
• These
sub-Numbers are smaller than the Number N, (the discriminant
W2 is outside the matter of N "W2 and in the form of N,).
2 Sub-Numbers N,lw3 where W3 is in the residue of N,. These sub-
Numbers are larger than the Number N, (the discriminant W3 is
in the residue of N, and outside the matter of N "W3)'
A sub-Number N "W2 of the first type will be called a low sub-
Number. A sub-Number of the second type will be called a high
sub-Number. The following diagram shows a low sub-Number and
a high sub-Number:
Number N, : • __ ..... , ...
Sub-Number N,/w2: •• ---... - ......
(low) I
Sub-Number N,/wa: •• _-.......J ______ ...
(high) Wa
Note that there are evidently as many lows as there are elements
in the form of N I (W2 must be in the form), and that there are as
many highs as there are elements in the residue of NI (W3 must be in
the residue).
The low set of Number NJ, denoted by Lo(Nd, is the set of low
sub-Numbers of N
• The symmetrical case (the set of high sub-
Numbers) is denoted by Hi(Nd, to be read 'high set of Number
14.7. The crucial point, then, is the following. Take a Number N,
its low set Lo(N) and its high set Hi(N). N is the one unique Number
of minimal matter to be situated 'between' the sets of Numbers which
are its high and its low sets.
This can be stated precisely as follows:
1 N is situated 'between' Lo(N) and Hi(N) in the sense that it is
larger than all the Numbers of one and smaller than all the
Numbers of the other.
2 All the other Numbers situated between Lo(N) and Hi(N) have a
greater matter than those of N. N is therefore the only Number
of minimal matter to occupy the interval between its low set and
its high set. Thus a Number N is a 'cut' between its low set and
its high set, a cut defined 'up to matter,.3 The two sets of sub-
Numbers Lo(N) and Hi(N) define N itself by way of location
(between the two) and material minimality.
14.8. The statement that N is between its high set and its low set is
quite trivial, since by definition all the low sub-Numbers are smaller
than N and all the high sub-Numbers are larger than N. The problem
is to establish that N is of minimal matter between the Numbers thus
situated, and that it is the only one to have this matter.
14.9. Principal lemma
Take NJ, a Number, and N
, another Number, smaller than NI and
of lesser matter than NI (so that N2 < NI and M(N
)). Then
either N2 is a Number from the low set of NJ, or there exists a
Number from the low set of NI situated between N2 and N
Let w be the discriminant of N I and N
. Since we suppose the
matter of N2 to be lower than that of N J, and since N2 < N J, w is
necessarily in the form of NI (it cannot be in the residue of N2 and
outside the matter of NJ, because then it would be in the matter of
N2 and outside the matter of N J, which possibility is excluded by the
fact that M(N
) < M(Nd). Consider the sub-Number N1/w. Since w
is in the form of N" it is a sub-Number from the low set of N, (it is
smaller than N d.
Up to, but excluding, w, N2 and N, are identical. If the discrimi-
nant w is outside the matter of N2 and therefore equal to its matter,
N2 is none other than the sub-Number Ndw, and is therefore a sub-
Number from the low set of N,. If w is in the residue of N
, then N2
is smaller than the sub-Number N ,Iw, because the discriminant of N2
and Ndw is necessarily w - N2 being identical to N, up to the ordinal
w (exclusive), and therefore also identical to N ,Iw, which is a parti-
tion at w of N" up to w (exclusive). Now, w is outside the matter
of N ,Iw, so we must suppose that it is in the residue of N
. So N2 <
Thus it is "established that N2 is indeed either a Number from the
low set of N, or smaller than a Number from the low set of N,.
14.10. An absolutely symmetrical chain of reasoning would prove
that, if N, < N2 and N2 is of a lesser matter than N" then either N2
is a Number from the high set of N" or else there exists a Number
from the high set of N, situated between N, and N
14.11. Conclusion: for every number lower than (or, respectively,
higher than) N, and of lesser matter than N" it is the case either that
it is a Number from the low set (or, respectively, the high set) of N"
or else that a Number from the low set (or high set) can be interca-
lated between it and N,. It is therefore impossible for any of these
numbers to be situated 'between' Lo(Nd and Hi(Nd (to be higher
than every element of Lo and lower than every element of Hi) whilst
at the same time being of lesser matter than N,. The result is that N"
which is indeed situated between its low set and its high set, is of
minimal matter with regard to all Numbers thus situated.
14.12. We will now demonstrate that NJ is the only Number of
minimal matter situated between its low set and its high set.
Suppose there existed another Number N
, situated between the
low set and high set of N" and of the same matter as N ,. Such a
Number could be represented as follows (with some abuse of our
Since N2 is of the same matter as N" the discriminant w of N, and
N2 is necessarily in the residue of N2 and in the form of N,. This
means that the sub-Number Ndw is in the low set of N,. Now this
sub-Number, N,lw, is manifestly larger than N2 (their discriminant,
once again, is w, which is in the residue of N2 and outside the matter
of Ndw). Thus it cannot be the case that N2 is larger than every
Number in the low set of N
H we had the arrangement:
- we could demonstrate in the same way that there must exist a
Number from the high set of NI which is smaller than N
(a good
exercise) ..
It follows that N, really is the only Number of minimal matter to
be situated between Lo(Nd and Hi(Nd.
NI is identified, 'up to matter' - as the unique minimal element of
that matter, once the 'between' position has been fixed - by the cut
of two sets of Numbers, the low set and the high set. We shall write:
N, = Lo(Nd/Hi(Nd. We shall call the cut Lo(Nd/Hi(Nd the canoni-
cal presentation of N I'
14.13. A remarkable characteristic of the canonical representation
of NI is that all the elements of Lo and of Hi are sub-Numbers of
. Every number can be represented on the basis of Numbers
deducted from lesser matters than their own.
The canonical presentation is a framing
of Number from above
and below, realised by means of more tightly controlled sections than
those carried out by Number.
Every Number is a cut within sets of sub-Numbers, every Number
operates at the limit of two series of Numbers subordinate and imma-
nent to it.
With this, the structuralisation of the concept of Number is com-
plete. Not only can a Number be located as a section cut from natural
multiplicities, but this section can itself be presented as a point of
cutting between two series of sections of the same type. A Number
is precisely thinkable as the hinge of its sub-Numbers. Number, so
far from being a simple entity, answers to theorems of decomposition:
it is a structure localisable in thought as a point of articulation of its
A Number exhibits, as a one-result, its immanent numerical
Cuts: The Fundamental
15.1. And so, let us penetrate into the swarming of Numbers.
A first remark, concerning what might be called the number of
Numbers: this number is precisely not a Number, it is not even a
consistent multiplicity. Numbers are numberless.
In fact, given that a Number is the pair of an ordinal and of a part
of that ordinal, not only are there at least as many different Numbers
as there are different ordinals, but there are many more, even if this
'more' flickers beyond the frontiers of sense. For each ordinal, there
are as many different Numbers as there are different parts of that
ordinal: if W is an ordinal, serving as the matter of certain Numbers,
there will be p(W) (the set of parts of W) forms - each one virtually
extractable by means of a numerical section from this matter.
Now we already know that the ordinals do not constitute a set.
'All' the ordinals cannot be counted for one in a set-theoretical recol-
lection. In other words, the ordinals form an inconsistent multiplicity.
Consequently, the same goes for Numbers.
But, what is more, for any given multiple whatsoever, we cannot
know exactly what the quantity of the set of its parts is. Certainly,
we know (Cantor's theorem) that it must be larger than that of the
initial set: it is always the case that P(W) > W. But 'how much' larger?
It has been proven (by Godel and Cohen's theorems) that the amount
of this excess is undecidable on the basis of the fundamental axioms
of set theory. In fact it is coherent within these axioms to say that
p(W) is 'immensely' larger than W; and it is also coherent to say that
it is 'minimally' larger.'
Ultimately, for every ordinal, there are always more possible
Numbers for which it is the matter than there are elements of the
ordinal itself. And, as things currently stand, the extent of this 'more'
can only be decided. Which is to say that the number of Numbers is
an inconsistency of inconsistencies.
The simplest way to put it is to say: Number is coextensive with
Being. It inconsists, is disseminated and profused just like the pure
multiple, the general form of being qua being.
15.2. This inconsistent swarming of Numbers gives us to anticipate
the difficulties that arise with regard to the identification of a specific
Number picked 'from the crowd'. Every Number is cemented into
the throng of those that pack in tightly, on its right (Numbers larger
than it) and on its left (smaller Numbers). No Number simply,
uncomplicatedly succeeds any other. Every microzone of the numeri-
cal domain teems with a numberless horde of Numbers. The numeri-
cal topology is peculiarly dense. And this is the problem: is it possible
to identify a Number as opposed to sets of Numbers? Or must we
consign ourselves, when we consider series of Numbers, infinite sets
of Numbers, to being unable to attach to them, univocally, any spe-
cific Number? Does the numberless throng of Numbers necessarily
lead us into 'those indefinite regions of the swell where all reality is
This is where trans-numeric inconsistency summons us to think
the cut. Is it possible, in a fabric so dense that nothing any longer
numbers it, to cut at a specific point? Can one determine, by cutting,
a singular Number?
15.3. This problem is not in the least bit academic, nor is it relevant
solely to the thinking of Number. We are told every day how
'the complexity of modern society' prevents us from making any cut,
any intervention. Contemporary conservatism no longer argues from
the sacredness of the established order, but from its density. Every
local cut, it says, is really a 'tear in the social fabric'. Leave natural
laws (the market, appetite, domination) to operate - because it is
impossible to interrupt them at any point. Every point is too depen-
dent on all the others to permit the precision of an interrupting
Thinking the cut in the hyper-dense, closely knitted fabric of
Numbers will allow us to conclude that such arguments are falla-
cious. Every point separates dense sets of Numbers, every Number is
the place of a cut, and, conversely, every cut prescribes one Number
and one only. Not 'indefinite regions', but 'a Constellation'.3
15.4. This problem also has a complex philosophical genealogy: that
of the dialectic between continuous magnitude and discrete magni-
tude. If the being of the continuum is grasped in its intimate coales-
cence, so that it is not constituted from distinguishable points, but
rather from complicated 'neighbourhoods', it must be thought as
disjoint from discrete quantity, which enumerates successive marks.
Up to, and including, Hegel, this opposition, which subsumes and
underwrites that between geometry and arithmetic, remains in the
position of an enigmatic real for the philosophy of quantity. In Kant,
still, it ultimately supports the duality of forms of sensibility: Space
is the transcendental figure of the continuous, Time - from which
proceeds number - that of discrete succession.
The most profound concept of the cut, a concept that plays an
immense role in modern thought,4 displaces and refounds the dialecti-
cal schema which considers the couplet discrete/continuous to be the
founding contradiction of the quantitative. This concept brings forth
a singularity - and therefore a basis for distinction - in the fabric of
the continuous, in the dense stuff of infinitely small neighbourhoods.
Overturning the customary order of thought, it shows how a certain
sort of interruption of the continuum defines a type of discreteness.
Rather than saying that the continuum is composed of points, it
determines points within the continuum, and even defines punctuality
on the basis of a cut in the continuum. The concept of cut substitutes,
for a problematic of composition, a problematic of completion: a
point comes to 'fill in' a juncture, or an imperceptible lacuna, in a
pre-given continuity.
15.5. Dedekind
invented the concept of the cut in order to define
irrational numbers.
He begins with rational numbers. We know that a positive rational
number is of the form .!!.., p and q being natural whole numbers. The
rational numbers provide our primary image of continuity owing to
the fact that their order is dense. A dense order is an order such that
between two ordered elements is always intercalated a third - and,
by reiteration of this property, an infinity of elements. If we take the
rational number 0 (which is rational because it can also be expressed
as any fraction %) and the rational number t, then 0 < t. But the
numbers t, t, t, etc. - and an infinity of numbers of the form .; -
intercalate themselves between 0 and t.
Density does not directly express a quantitative property: the ratio-
nal numbers are an infinity of the type belonging to the countable,
an infinity no greater than that of the natural whole numbers, and
the latter, being none other than the finite ordinals, do not present a
dense order: there is no natural whole number between nand n + 1.
Density is really a topological property of order: excluding the simple
idea of 'another step', of the well-determined follower of a term, it
proposes instead a sort of general coalescence in which every term
'sticks' to an infinity of neighbours. The density of an order is a
topological property, whereas succession is an algebraic property.
Density is 'quasi-continuous', one can approach a rational number
as closely as one wishes through other rationals. One even gets the
feeling that, between two rational numbers and, more generally,
between two terms of a dense order, there is no place for numbers,
or terms, of another type, since the whole interval, no matter how
small it is, is already populated with an infinity of rationals, or an
infinity of terms of the dense order.
Now, it is precisely in this quasi-continuity of rationals that
Dedekind will, by means of the cut, define additional 'points' that
will complete the apparently uncompletable density of the rationals
and obtain a 'true' continuum, through interruptions in their
We will return in greater detail to this procedure in chapter 16.
But schematically: Dedekind considers disjoint sets of rational
numbers RI and R
, for which every element of RI is less than every
element of R2 and which, R2 having no rational internal maximum
nor RI any rational internal minimum, are two 'open' sets, one high,
the other low. Dedekind then identifies a real number as occupying
the place of a cut between RI and R
• This real number will be both
the upper limit of RI and the lower limit of R
• The density of the
order of rationals plays an essential role in this construction, once it
is understood that density and the cut, far from being exclusive, are
paired together in thought.
It must be noted straightaway that this procedure seeks to define
real numbers, the rational numbers being supposed to be known. The
Dedekind cut is wholly an operation of completion: where there is
nothing, no rational number, the name of something 'extra' comes
forth. The real number defined by the cut R/R2 fills in that which,
thought purely from the point of view of rationals, is a void in the
density, and thus a void to which nothing attests. This is why the cut
founds a new species of numbers, which 'complete' the initial density
and retroactively indicate that this density was not so dense that gaps
could not be discovered therein.
15.6. We cannot hope to 'complete' the inconsistent domain of
Numbers, nor to found, outside Number, a hyper-number which
would name the invisible lacunae in it. Our Numbers are uncomplet-
able, being coextensive with Being (see 15.1). All the Numbers are
already there. What could a cut mean in such conditions?
Nevertheless, there is a very strong concept of the cut for Numbers.
This concept holds 'up to matter', like that of the singular element
separative of a Number and its sub-Numbers, of its identity as cut
between its low set and its high set (see chapter 14).
This concept of the cut is presented in the following theorem
which, articulating the inconsistent swarming of Numbers with the
precision and the unity of a punctual cut, well deserves the name of
fundamental theorem of the ontology of Number:
Given two sets of Numbers, denoted by B (for 'from below') and
A (for 'from above'), such that every Number of set B is smaller
than every Number of set A (in the order of Numbers, of course),
there always exists one unique Number N of minimal matter
situated 'between' Band A. 'Situated between' means that N
is larger than every element of B and smaller than every element
of A.
The Number N is evidently not the only one between Band A.
The numerical swarming is such, the density is so considerable, that
such a solitude would be unthinkable. But it is the only number to
be found with its matter. All the others have a larger matter, in a
rigorous sense, since matters are ordinals: the ordinal-matter of N is
minimal for the property 'is the matter of a Number situated between
the sets of Numbers B and A'.
It will not surprise us at all to find minimality here: it is a classic
organisational principle of ordinals. What is surprising is:
- that such a Number should exist;
- that it should be unique.
Its existence founds the principle of the cut. If two sets of Numbers
are like B and A (every Number of B being smaller than every
Number of A), then one can still speak of what exists 'between' B
and A and is neither of B nor of A, in spite of the prodigious density
of the order of Numbers. It is thus possible to make a cut in the
hyper-dense fabric of this order.
Uniqueness (up to matter, which is to say uniqueness of the
Number-<:ut of minimal matter) founds the principle of identification,
the persistence of the count-for-one even where all is coalescent, in
dense neighborhoods. A cut designates one Number, and designates
it on the basis of sets of Numbers. We will hold that no complexity,
even one pushed to the point of inconsistency, no density, even one
pushed to the finest infinitesimal proximities, can authorise the pro-
hibition against cutting at a point.
15.7. The rest of this chapter is dedicated to the demonstration of
the fundamental theorem, the only theorem in this book that is a
little complex.
I do not, moreover, intend to give all the details of the proof.
However, we are at the heart of the mathematics of Number, and
what must be put into play in order to think the cut is of a conceptual
interest far surpassing mathematical ontology. All truth-procedures
proceed via a cut, and here we have the abstract model of every
strategy of cutting. The intellectual effort demanded of the reader will
lead him or her, I am quite sure, to beatitude in the Spinozist sense.
15.8. Upper bound of a set of Numbers
Since we are engaged in investigations whose character is topological,
and since in particular we are wondering how to find Numbers larger
(or, respectively, smaller) than a given set of Numbers, let's begin
with the simplest concept, that of an upper bound: given a set of
Numbers, does it make sense to speak of a 'unique' Number larger
that all those in the set?
Once more we must, in view of the proliferation of Numbers, avail
ourselves of a concept 'up to matter'. We will prove the following:
if B is a set of Numbers, then there exists a Number N which is the
unique Number of minimal matter to be larger than all the Numbers
in set B. We will call this N the upper bound of B. Right away the
upper bound exhibits a surprising characteristic: it is always a Number
written (W,W) - that is, a Number whose form is its whole
15.9. Take B, a set of Numbers. Consider the ordinal defined as
follows: 'the smallest ordinal W such that, for every Number N of
set B, there exists a WI E W which is either in the residue or outside
the matter of N'.
One such ordinal W exists, because B is a set, and is therefore
consistent. If W did not exist, that would mean that all ordinals
would fall into the form of at least one Number N of B. But 'all
ordinals' is an inconsistent multiplicity, and consequently B would
also be an inconsistent multiplicity, and would not be able to be
thought as a set.
That there should exist such a 'smallest' W results from the prop-
erty of minimality that characterises the ordinals.
W having been specified, now consider the Number (W,W). This
number is larger than every Number in set B. In fact, by the definition
of W, for every Number N of B there exists a WI E W which is in its
residue or outside its matter. Now, as the form of (W,W) is W, every
WI E W is in the form of (W,W). The discriminant of a Number N
and of (W,W) is necessarily the smallest WI E W that is in the residue
or outside the matter of N. And, since this WI is in the form of (W,W),
the residue (or outside-matter)/form relation demands that (W,W)
should be larger than N.
Since a Number larger than every Number in B exists - namely
(W,W) - one of minimal matter must exist, in virtue of the ordinals'
property of minimality.
Therefore, let (WI,X) be a Number of minimal matter for the
property 'being the matter of a Number larger than all the Numbers
in B'. Its form X is in fact equal to WI.
For, if X differed from WI - if, that is, the form of the Number
was not its whole matter - that would mean that there existed at least
one ordinal W2 E W which was in the residue. Consider then the
sub-Number of (WI,X) obtained by partition at W2 - that is, the sub-
Number (WZ,X/W2). Since Wz is in the residue of (W"X), the sub-
Number (WZ,X/W2) is in the high set of (W I ,X) (see 14.7). It is there-
fore larger than (WI,X), and a fortiori larger than every Number in
B, since this is already the case for (W"X).
But that is impossible, because the number (WZ,X/W2) is of lesser
matter than the Number (WI'X). Now, we supposed that (WI,X)
was of minimal matter for Numbers higher than every number
in B.
Our initial hypothesis must be rejected: there does not exist in
(WI,X) any element that is in the residue, which is to say that the
form occupies the whole matter, and that the Number must be
written (W"Wd.
There exists therefore one Number only of minimal matter that is
higher than all the Numbers of set B: it is the Number (W"Wd, where
W I is this minimal matter.
We can thus legitimately speak of the upper bound of a set of
Numbers. Already the theme of unicity comes to inscribe itself as bar,
or caesura, in the hyper-dense swarming of Numbers.
15.10. Lower bound of a set of Numbers
Reasoning totally symmetrical with that employed for the upper
bound will permit us to define the unique Number of minimal matter
that is smaller than a set A of Numbers. This will be the lower bound
of the set A. We will see that, this time, this Number is written (W
its form is void, the numerical section does not extract anything from
the matter W
Let A be a set of Numbers, and let W be the ordinal defined thus:
'the smallest ordinal such that, for every Number N of A, there exists
a WI E W which is either in the form, or outside the matter of N'.
This ordinal exists necessarily, because A is a set, and in virtue of the
principle of minimality (see above).
The Number (W,O), whose matter is Wand whose form is the
void, is smaller than every Number of A. In fact, all ordinals WI E
Ware in the residue of (W,O). Now, for every Number N of A, by
the definition of W, there exists a WI E W which is either in the
form of N, or outside the matter of N. The smallest such WI is the
discriminant of N and of (W,O), and its location means that (W,O)
Hence there exists a Number smaller than every Number in A, and
- by the principle of minimality - there exists at least one of minimal
matter, say (W
It is easy to prove that X is necessarily the empty set. If it were
not, that would mean that there existed a W3 E W 2 which was in
the form of (Wz,X). But then the sub-Number of (Wz,X) obtained
by partition at W3, that is, (W3,Xlw3), would be in the low set of
(Wz,X) (see 14.7). It would then be smaller than (Wz,X), and therefore
smaller than every Number in A, although of lesser matter than
(Wz,X): which is impossible in view of the minimality of Wz for this
Therefore, there exists one unique Number of minimal matter that
is smaller than every Number in A. It is the Number (Wz,O), where
is this minimal matter. The Number (W
,0) is the lower bound
(up to matter) of set A.
15.11. Fundamental theorem, first part: Existence
'Existence' means here: existence of at least one Number situated
between two sets of Numbers B and A, which, in an abuse of our
usual notation, we shall generally write as B < N < A.
Take B and A, two sets of Numbers such that every Number of B
is smaller than every Number of A. Our technique will consist in
constructing, between B and A, step by step - that is to say, ordinal
by ordinal - starting from 0, a Number N 'suspended' at every step
in such a way as to assure us that nothing up to the ordinal W in
question, which is to say, for every step taken in the procedure - can
force the Number N to be smaller than a Number of B, or larger than
a Number of A. We might also say that we are going to construct
N from its sub-Numbers of intersecting matter, by 'choosing' to put
an ordinal W in the form or in the residue of the Number N under
construction, depending on the relationship between the segment of
the procedure of N which goes from ° to W, and the various sub-
Numbers in B and in A.
The underlying idea is that the construction of a cut makes nec-
essary a local domination of the substructures implicated in the
course of this construction. This is a general law of practice, at
least in so far as the latter aims at effects of cutting (foundational
This technique boasts the very great interest of highlighting the
link between cutting and a sort of procedure of neutralisation. So
that N can slide in between the Numbers of B and the Numbers of
A, we are going to remain mindful of the fact that the principle of
order, at every point of N, 'neutralises' the discrimination between
the Numbers of B and the Numbers of A. The great difficulty being
to know when to stop ourselves, when to fix the matter of the
Number N, which we would have traversed, all the while postponing
its closure.
In all domains of thought, to proceed with a precise cut in a
densely ordered fabric is to calculate a prudent tactics of inser-
tion step by step, and then to risk a stopping point which will
irreversibly fix the intermediary term. The cut thus combines the
neutrality of the interval and the abruptness of the interruption.
This is why great strategies of thought must always attain a
mastery both of the patience which, point by point, opens and
enlarges a lacuna, and of the impatience which comes to seal and to
name its existence from this moment forward, without return or
15.12. So we begin from the ordinal 0, and we traverse the ordinals,
assigning to each a value f(W) - the values being F (for form), R (for
residue), or M (for matter). The value M can obviously only be given
once, and last of all, because the Number N that we want to construct
has only one matter. For an ordinal W, if f(W) = F, we will put Win
the form of the Number N under construction; if f(W) = R, we will
put it in the residue. So long as we have not assigned the value M,
the sub-Numbers are still 'under construction'. The procedure
amounts to fixing a location-status for each ordinal W, so that the
sub-Number NIW, as the procedure continues, will appear retroac-
tively as never constraining N to be larger than any Number of A,
or smaller than any Number of B.
The strategic patience of the construction of a cut consists in insert-
ing additional local values without compromising the chances of a
global cut. It is a work that proceeds point by point, but is retroac-
tively decided as an irreversible and general caesura.
We will denote by Nb and Na, with indices if need be, the Numbers
of Band A. Ni will designate the intervallic Number, the Number we
wish to construct between Band A.
If, for a given Number Nb (or respectively Na), the values attrib-
uted by f to ordinals smaller than an ordinal W (values of the type
F or R, which the ordinals of the Ni under construction take) are
exactly those which locate these ordinals in Nb (or respectively in
Na), then we say that W identifies Ni and Nb (or respectively Na).
W's identifying Nb (or Na) and Ni means that, in every case, no
ordinal smaller than W can discriminate between Nb (or Na) and Ni.
In particular, the discriminant of Nb, or Na, and the segment of Ni
under construction (a segment which ranges from 0 to W exclusive)
cannot figure in the ordinals inferior to W. Which amounts to saying
- and this is the most tractable form of the relation of identification
at ordinal point W - that, up to W, the 'sub-Number' Ni/W is identi-
cal to the sub-Number Nb/W (respectively Na/W).
We will denote by Id.(W,Nb) the fact that W identifies Ni and Nb.
And the same thing for Na. All the while we should keep in mind
that Id.(W,Nb) means that Ni/w = Nb/W.
The strategic idea is to construct an Ni 'neutralised' for order, by
making sure, each time one comes to 'the end' of a series of ordinals
which identify Nb (or respectively Na) and the Ni under construction,
that the choice of a value for f(W) will not be able to compromise
our chances of positing a hypothetically completed Ni, which would
be intervallic between Band A. We must just make sure that no
ordinal comes to be in the position of an unfavourable discriminant
forcing Ni to be smaller than a Number of B, or larger than one of
A. The prudence of the cut consists here in never risking losing the
chance to take up an intervallic position. Conserve its chances, that
is the maxim of the 'step by step' phase of the construction of
a cut.
15.13. We will posit the following rules - rules of construction of Ni
for the ordinals starting from 0:
RULE 1: If Id.(W,Nb) and W is the matter of Nb, then f(W) = F.
We put the ordinal W in the form of Ni whenever, at the end of
an Nb/W identical to Ni/w, W is the matter of Nb. So, using a
black square to denote a belonging to the form:
Ni "'"
RULE 2: If Id.(W,Na) and W is the matter of Na, then f(W) = R.
The diagram should be clear, marking with I a belonging to the
Ni ~
NilW vW
N a ~
RULE 3: If rules 1 and 2 do not apply for a given W, but instead
we have an Nb such that Id.(W,Nb) with W in the form of Nb,
then f(W) = F. If cases 1 and 2 do not apply, we put W in the form
of Ni each time that, at the end of an NblW identical to NilW,
W is in the form of Nb:
Ni ___ --------------__ ~ ~ .
RULE 4: If rules 1 and 2 do not apply, and we have an Na such
that Id.(W,Na) with W in the residue of Na, then f(W) = R:
~ ..
l/ ..
RULE 5: If none of the first four rules apply, it must be the
case that, for the W considered, no Nb such that Id,(W,Nb) has
W for matter or in its form, and that no Na such that Id,(W,Na)
has W for matter or in its residue. Under such conditions, at
point W, if there exists an Nb for which Id.(W,Nb), W is in
the residue of Nb, and if it is the case that Id.(W,Na), W is in
the form of Na. We then say that f(W) = M, which completes
the construction of Ni.
As justification for this rule, note the following: since all Nb (or
respectively Na) where W is not thus located - so, W in the residue
(or respectively in the form) - are such that W does not identify
them with Ni, then, for these Nb (or Na), it is the case that NbfW...;:.
NifW (or respectively NafW ...;:. NifW). In other words, these Nb and
these Na have already been discriminated, before ordinal W, by the
process Ni. The only Nb and Na not to have already been discrimi-
nated are those where W is in the residue (or respectively in the
Given this remark, we can state that rule 5 prescribes witb com-
plete justification tbe decision of closure of the process Ni. We can
posit: f(W) = M, thereby fixing W as the ordinal-matter of Ni, and
therefore as that place where the process of the construction of Ni
If W is the matter of Ni, it is located outside the matter for that
Ni supposed closed in W. Now, W does not discriminate Ni from
Nb where W is in the residue, or from Na where W is in the form.
The location for Ni will remain 'between' B and A, since the schema
of the order-relation is precisely R < oM < F. We will have:
Nb -
":: ~
I d ~ m
Na .!:
Closure is entirely possible, since, beyond ordinal W, all Nb and
Na are discriminated by Ni (before W through rules 1 to 4, at point
W by rule 5); and our rules reflect the fact that this discrimination
always goes in the direction Nb < Ni < Na.
This regulation, however, merits immediate examination.
15.14. It is essential to confirm that our rules do not contradict one
Take for example rules 1 and 2. If by some mischance it should
happen that at the same time Id.(W,Nb) and Id.(W,Na), with W the
matter both of Nb and of Na, then W would have to be placed
simultaneously in the form and in the residue of Ni ...
But such a case cannot arise. Because, if W is the matter of Nb
and of Na, since every Number of B is smaller than every Number
of A, it is the case that Nb < Na. And, since they have the same matter
W, their discriminant must be less than W, which is to say that there
is at least one ordinal WI E W which doesn't have the same location
in Nb and in Na. It is therefore not possible for sub-Numbers NblW
and NalW to be identical. This means, moreover, that, if both
Id.(W,Nb) and Id.(W,Ba), their common identity must be NilW. So
rules 1 and 2 are compatible.
But take rules 3 and 4. If by some mischance there is a W for which
rules 1 and 2 do not apply, and there exist Nb and Na for which,
firstly, Id.(W,Nb) and Id.(W,Na), and, secondly, W is in the form of
Nb and W is in the residue of Na, W would have to be placed both
in the form and in the residue of Ni.
But of course such an unfortunate circumstance cannot arise.
Because, if W is in the residue of Na and in the form of Nb, then it
discriminates between Nb and Na. But this could not be their dis-
criminant, otherwise it would be the case, with regard to this loca-
tion, that Na < Nb, which is prohibited by B < A. Therefore the
discriminant is smaller than W, and, as before, it is impossible that
NblW = NalW; which makes it necessary to suppose their common
equality to NilW.
15.15. Now we will see whether, with these rules, we do indeed
preserve our chances that Ni will slip in between all the Numbers
of B and all the Numbers of A, and therefore between all Nb and
all Na.
When we apply rule 1, we give the value F to the ordinal W. This
certainly cannot make Ni become less than a Number of B, because,
if W is the discriminant of Ni and of an Nb, being in the form of Ni,
it will always be the case that Nb < Ni.
But, given the fact that we put W in its form, don't we risk Ni
becoming larger than a Number of A? For this it would have to be
the case that W was the discriminant of Ni and of an Na. But then
it would also ultimately be the discriminant of the Nb of which W
is the matter (since we apply rule 1) and of Na. Now, we know that
Nb < Na. If their discriminant is the matter of Nb, it must be in the
form of Na. This location of W - W being the discriminant of Ni
and Na - prohibits us from having the order Na < Ni.
So, in applying rule 1, we can be sure that the location that we fix
for W in the Number Ni under construction entails neither an unwel-
come and frustrating Ni < Nb, nor a fatal Na < Ni. At point W, Ni
stays situated 'between' Band A.
The examination of the other rules leads us to the same conclusion.
Let's carry out this examination for rule 5 (for rules 2, 3 and 4 the
methods are the same as for rule 1. Let the reader prove this as an
exercise, with the help of the note,9 and above all of the diagram
Rule 5 comes into play when rules 1 to 4 are not applicable. The
W under consideration makes no identification between any Nb (or
Na) and Ni if W is located as matter of Nb (rule 1), matter of Na
(rule 2), form of Nb (rule 3) or residue of Na (rule 4). If, then, it is
the case that Id.(W,Nb), or Id.(W,Na), it is because W is in the residue
of Nb and/or in the form of Na. These two hypotheses are compatible
this time: the identifications in question could obtain, and W could
be both in the residue of Nb and in the form of Na. Rule 5 then
compels us to make the gesture of closure f(W) = M, which deter-
mines W as matter of the intervallic Number Ni. In the Ni thus
closed, W is located outside the matter. Can this choice make Ni less
than some Na, according to the relation R < oM? No, because, if W
discriminates between this Na and Ni, with W in the residue of Na,
this would be a case for the application of rule 4, which would
exclude the use of rule 5. And, in the same way, it cannot be the case
that Ni < Nb according to the relation oM < F, because the location
of the discriminant W in the form of an Nb compels, for W, the use
of rule 3 rather than rule 5. Rule 5, applied when it is proper to do
so, cannot entail that Ni < Nb. And, as it cannot entail Na < Ni
either, it leaves the procedure Ni, at point W, in the interval between
Band A.
So it is that, at every ordinal point W, the application of our rules
'locally' situates Ni, in the form of the sub-Numbers NilW, in an
intervallic position with regard to Band A. Our step-by-step labour
is pursued without Ni surpassing any Na, or being surpassed by any
Nb. We conserve our chances all the way through the construction.
An enlarged diagram shows how Ni proceeds. We have, above, some
Numbers Nb of B, below, some Numbers Na of A, and, in the middle,
the process of Ni. The ordinals WI to Ws present, in order, cases of
the application of the five rules. Squares, asterisks and bars designate
form, matter and residue. You will recall that, when a point is marked
in an Nb or an Na, it means that, before that ordinal point, Na (or
Nb) is identical to Ni (relation of identification at an ordinal
1/ 1/ ..
/ /
.. Ni--'
rule 1 rule 2 rule 3 rule 4 rule 5
...... ~
if fules 1 and 2 do not apply if fules 1 and 4 do not apply
The whole subtlety of the enterprise lies in minimising the risks,
in making sure not to increase the value of Ni to point W (in particu-
lar, in not giving it value F) until one is sure that this increase will
have no effect with regard to A; and in not decreasing this value (the
value R) unless all effect with regard to B is excluded. Thus Ni, per-
petually maximising the neutralisation of the effects of order, slips
in between Band A.
And, when the time for closure arrives (rule 5), for a W situated
between residue (Nb) and form (Na), we retroactively set the seal on
the tactics, arriving at a Number globally situated between B and A,
because it is protected, locally, from any prohibition against this
15.16. Fundamental theorem, second part: Unicity
We have just indicated the strategy - combining local, neutralising
patience with a global decision of closure - that allows the existence
to be established, in every case, of at least one Number situated
between two sets of Numbers B and A such that (in an abuse
of notation) B < A. In virtue of the principle of minimality of
ordinals, there must exist at least one such Number of minimal
matter: we wiII consider the property 'being the matter of a Number
situated between B and A', and the minimal ordinal for this
It remains to be shown that a Number of minimal matter situated
between B and A is unique, which will permit us to identify the
numerical cut between Band A.
Suppose that there were two: we would have the following
- with N 1 and N
being of the same matter (minimal for this
Since Nl and N
are of the same matter, Nl < N
means that the
discriminant must be in the residue of Nl and in the form of N
• Take
this discriminant, w. Consider the sub-Number N1/w of N
• Since w
is in the residue of N h this sub-Number belongs to the high set of
: it is therefore larger than N
• But, since w is the discriminant of
N I and N
, and therefore the smallest ordinal to discriminate between
them, then Nl and N
are identical up to w (exclusive). This means
that the sub-Number Ndw is identical to the sub-Number Nz/w. The
discriminant of Ndw and of N
can only be w, which is outside the
matter of N1/w and in the form of N
• Consequently, N1/w < N
So finally, we have the arrangement:
Which is to say that N dw is also situated between Band A. But this
is impossible, given that it is of lesser matter than that of N J, which
is supposedly minimal for the location between Band A.
We must reject our initial hypothesis: there are not two Numbers
of minimal matter between B and A, there is only one.
The two sets B and A therefore determine univocally one Number
of minimal matter situated between them. This Number will be called
the cut of B and A, and we will posit that N = B/ A, each time that
N can be identified as the unique cut of B and of A.
15.17. There is one very peculiar case of the cut: take two Numbers
N 1 and N
such that N 1 < N
• And take, for B and A, the sets which
have for elements only N I and only N
, that is, the singletons (N d
and (N
). We remain within the parameters of the fundamental
theorem, which is to say that there exists a unique Number N3 of
minimal matter situated between Nl and N
• We thus rediscover here
the classic condition of density of an order, which we have mentioned
with regard to the rationals: between two Numbers there always
exists a third, and thus an infinity of Numbers. For us, besides this,
there is an additional determination: between N I and N2 there is
always a unique Number of minimal matter.
We can therefore put forward a principle which everything gives
us to expect, and of which the unicity of the cut provides the infinitely
strong concept: the order of Numbers is dense.
But more profound than this is the correlation in thought between
this numberless density, this coalescence which inconsists in the
approach to all Number, and the possibility of counting for one the
Number of minimal matter which cuts the fabric without lacuna of
numericality at a certain point.
'Cut' here designates the incision of thought in the inconsistent
fabric of being, that which Number sections from the ground of
Nature. It is a concept of singularity. Perhaps the concept of singular-
ity, at least in the order of being. For there is that other singularity
which cuts across being, and which is the event.
The Numberless Enchantment
of the Place of Number
16.1. A review, to begin with.
1 A Number is an ordinal - the matter of the Number, M(N), in
which is sectioned a part of that ordinal- the form of the Number,
F(N). We also consider that part of the ordinal-matter that falls
outside the section, outside the form: the residue of the Number,
2 The location of an ordinal with regard to a Number N is its posi-
tion in (belonging or non-belonging to) one of the three 'compo-
nents' of Number: form, residue, matter. There are three locations:
in the form, in the residue and outside the matter.
3 The discriminant of two Numbers Nt and N2 is the smallest
ordinal not to be located similarly in both Numbers. If no
such discriminant exists, then the two Numbers are equal (they
have the same matter, the same form, and therefore the same
4 Depending on the location of the discriminant, we can define an
order-relation (transitive and non-reflexive) between two different
Numbers. We denote this through Nt < N2 and by saying that Nt
is smaller than N
• This relation is a total order over the domain
of Numbers in the sense that, given two different numbers Nt and
N2, it is always the case either that Nt < N2 or N2 < Nt.
5 The order-relation is dense: given two Numbers Nt and N
Nt < N
, there always exists an N3 which comes in between Nt
and N2: Nt < N3 < N2.
6 Take a Number NI of matter WI and an ordinal WI smaller than
WI (so that WI E Wd. The Number of matter Wb which is exactly
like NI up to WI exclusive (the form of this Number being con-
stituted by all ordinals smaller than WI that are in the form of
Nd will be called a sub-Number of Nb a sub-Number of NI
which is a 'cut' of NI at point WI. We denote this sub-Number
7 Amongst the sub-Numbers of Nb some are smaller than NI (when
WI is in the form of Nd, others are larger than NI (when WI is in
the residue of Nd. The former, gathered together, constitute the
low set of Nb denoted by Lo(Nd. The latter constitute the high
set of Nb denoted by Hi(Nd.
8 It can be proved that N I is the cut of its low set and its high set
in the following way: it is the Number of minimal matter situated,
according to the order of Numbers, between the low set and the
high set (larger than every Number in the low set and smaller
than every Number in the high set).
9 More generally, it can be shown that, given two sets of Numbers
such that all those of the first set are smaller than all those of the
second, there exists a unique Number N of minimal matter situ-
ated between these two sets. Taking two such sets B and A, we
can say that this Number N is the cut of B and A, which is written
N = BfA. Thus NI = Lo(NdfHi(Nd. This specified cut is called
the canonical presentation of N I.
16.2. We will now take a stroll through the borderless domain of
Numbers, pointing out some of them, and in particular all those tra-
ditional species: natural whole numbers, negative whole numbers,
ordinals, rationals, reals. But also so many others, which finitude and
the wretchedness of our inherited practice of Number keeps from us.
How negligible are numbers amongst Numbers! The being of Number
exceeds in every direction that which we know how to negotiate. Our
strength, however, is that we possess a way of thinking of this excess
of being over thought.
16.3. Zero
There is a very distinctive Number, the Number (O,O), whose matter
is the void, and whose form, consequently, is also the void. This
Number inscribes as numerical gesture the absence of every gesture,
in default of any matter. It is absolute Zero, the Number without
numericality. Of course, its ontological foundation is the empty set,
the suture to being of every text, the advent of being qua being
to the thinkable. There is no doubt that it is this void that we are
thinking here as Number. But thinking it as Number makes a differ-
ence. It is not for example the same thing, not the same Number, as
it would be if the void was only in the position of matter, or only in
the position of form. The number ((0),0), or (1,0), whose form is
void, cannot at all be identified as the Zero of Number. Certainly, the
act of sectioning it is equally null, it doesn't extract anything from
its matter, but this matter suhsists unaltered, constituting, in the
absence of any act, the real substance of that which this gesture never
even started. The only true Zero is that which subtracts itself from
all numerical gesture because it has nothing, no material or natural
multiplicity, upon which this gesture could be carried out or not
carried out. Zero is thus outside all appreciation, positive or negative,
of the act of numerical section. It is, very precisely, neither positive
or negative. It subsists in itself, inaccessible to all evaluable action.
Zero is being qua being thought as Number, from within ontology.
16.4. Since we have said, a little metaphorically, that Zero is neither
positive nor negative, can we not give a precise numerical sense for
these adjectives? Elementary arithmetic already introduces - to the
obscure relish of every schoolchild - whole negative numbers such
as -4.
Consider for example the Number N, whose matter is the limit
ordinal co, and whose form has only the ordinal ° as element. Which
is to say that the form is the singleton of 0, and that the number NJ
can be written: (co,(O)). If we compare this Numher to Zero, that is,
to (0,0), we can clearly see that their discriminant is 0, which is in
the form of N, and outside the matter of Zero (any ordinal whatso-
ever, including 0, is outside the matter of Zero, which has no matter).
The rules of order indicate to us then that N, is larger than Zero.
It makes sense to say that N, is positive.
Consider now the Number N
, whose matter is also the limit
ordinal co, but whose form is this time the singleton of 1. This
Number N
can be written (co,(l)). Once again, the discriminant of
and Zero is 0. It can be found this time in the residue of Nh since
the form of N
does not contain ° (it only contains 1), but its matter,
co, does contain it, co being the limit collection of all the finite ordinals,
including ° of course. We can see, then, that 0, being outside the
matter of Zero and in the residue of N
, is smaller than Zero.
So it makes sense to say that N
is negative.
16.5. Positive Numbers and negative numbers
Our examples can be generalised in the following fashion: the dis-
criminant between Zero and any other Number whatsoever is always
the empty set O. For Zero is the only number whose matter is void,
and therefore the only Number where 0 is located outside the matter.
For every other Number, 0 is located in the form or in the residue.
And, since 0 is the smallest ordinal, it is certainly the discriminant of
Zero and of every number other than Zero.
The situation is very simple, then: if any Number other than Zero
has 0 in its form, then it is larger than Zero. If, on the other hand,
o is in its residue, it is smaller than Zero, since 0 will always be
outside the matter of Zero.
We will thus define positive and negative Numbers in the following
way: A Number is positive if 0 is an element of its form. It is negative
if 0 is an element of its residue.
16.6. Some significant consequences of the definition of positive and
negative Numbers:
1 Since Zero is without matter, without form and without residue,
o cannot be an element either of the form or of the residue of
Zero. The description in 16.3 is thus transformed into a mathe-
matical concept: Zero is neither positive nor negative.
2 Zero is not at all the smallest Number. It is larger than every
negative Number, and negative Numbers constitute, to all appear-
ances, a limitless, inconsistent domain. Between the negative
Numbers and the positive Numbers, Zero lies at the centre of that
which has no periphery.
3 Zero is not defined by extrinsic operations, it is not introduced as
the 'first' term of a succession, nor as the 'neutral element' of an
operation (an attribute which it possesses incidentally and second-
arily). It is characterised by its numerical being. We have not
strayed from our ontological path, which subordinates all opera-
tional or algebraic considerations to immanent characterisation.
4 More generally speaking, the categories 'positive' and 'negative'
have been introduced into the consideration of the order of
Numbers only for convenience of exposition. The predicate 'has
o in its form' or 'has 0 in its residue' are wholly intrinsic. The
examination of the being of a Number alone tells us whether it
is positive or negative, without comparing it with any other
S Positivity does not depend in the least upon the 'quantity'
of the matter of a Number, or the size of its form, but only
upon the location of the void. The Number (2,(0)) is positive,
whilst the Number (00,(00 - 0)), whose matter is 00 and whose
form takes in all of this matter apart from 0, is negative. There
is finite positive numericality, and infinite negative numericality,
regardless of whether the question is one of matter or one
of form.
6 If a Number N is positive, then, since 0 is in its form and is neces-
sarily minimal, it follows that every sub-Number Nlw of N (except
for Zero, which is a sub-Number of every Number, the sub-
Number N/O) is also positive: the elements of the form of Nlw
are actually the elements of the form of N up to the ordinal w,
and, unless w is 0, 0 will be amongst these elements, since N is
positive. Similarly, every sub-Number of a negative Number N,
apart from 0, is negative (it has 0 in its residue, as N does). In
particular, the non-null elements of the low set and all the ele-
ments of the high set of a positive Number are positive; likewise,
all the elements of the low set and all the non-null elements of
the high set of a negative Number are negative.
16.7. Meditation on the negative
The concept of negativity, as proposed by the universe of Numbers,
is every bit as profound as its apparent paradoxicality suggests. One
might think at first that negativity consisted precisely in the incorpo-
ration of the void into the form of Number. Isn't there more positivity
in a form that has not been marked by the stigma of nothingness?
Isn't the plenitude of the numerical section better assured if it expels
from its positive production that dubious index of the multiple that
allows no presentation?
Number enjoins us here to disabuse ourselves of any remaining
temptation towards an ontology of Presence. If the lack of void in
the form of Number seems 'positive', this is the case only if we
identify being with the plenitude of the effectively presented. We
are then tempted to index to the negative every occurrence of that
which presents nothing, every mark whose multiple-referent is sub-
tracted. But the truth is entirely otherwise: it is precisely under this
mark that being qua being comes to thought. In which case there
is less ontological dignity in a Number that does not retain this
mark in its form than in a Number that does so retain it. It is from
the point of the void that the dignity of being, the superiority of a
Number, can legitimately be measured. Numerical superiority is the
symbol of this superiority with regard to what is at the disposal of
The ontological clarity (for a subtractive ontology) of the state-
ment 'a Number is negative if the mark of the void is in its residue'
underlies what might be called the ethical verdict of Number. I hope
to show one day that what is Evil, in any situation where the void is
attested to (and such, singularly, are post-evental situations), is the
treating of that testimony precisely as if it were a residue of the situ-
ation. What is Evil is to take the void, which is the very being of the
situation, for unformed. The forms of Evil declare substance full and
luminous, they expel every mark of the void, they rusticate, deport,
chase off, exterminate those marks. But the verdict of Number tells
us: it is in this claim to full substance, in this persecution of the occur-
rences of the void, that resides, precisely, the negative. A contrario,
positivity assembles and harbours the marking of the void within its
forms. And, this being so, it accords thought to being in an intrinsi-
cally superior fashion.
To take the void for a residue is a negative operation, a detestable
'purification'. Every true politics, in fidelity to some popular event,
takes on the guardianship of the void - of that which is un presented,
not counted, in the situation - as its highest duty in thought and in
action. Every poem seeks to uncover and to carry to the formal limits
of language the latent void of sensible referents. Every science treats
positively the residue of its own history, that which has been left
outside of its form, because it knows that precisely there dwells that
which will refound and reformulate its system of statements. All love
ultimately establishes itself in the joy of the empty space of the Two
of the sexes which it founds, and from this point of view the romantic
idea of a full, fusional love, under the purified sign of the One, is
precisely the Evil of love.
The negative, as its concept is established by Number, is a punctual
discord of thought and of being. 'Negative' is every enterprise of
formation which abandons, fails to cherish, this unique point upon
whose basis there can be forms and the unformed, forms and resi-
dues; the point where being, in the guise of the un presented, assures
us that we do not think in vain.
16.8. The symmetric counterpart of a Number
Not much needs to be done in order to 'negativise' a positive Number:
it suffices to remove 0 from its form. Number teaches us the precarity
of the positive, its a-substantial character. It is at the mercy of the
transfer of one single point to the residue. And this point is the most
transparent of all, that point that is not supported by any multiple-
presentation: the mark of the void.
This idea of the transfer of a term from one location (here, the
form) to the 'opposite' location (here, the residue) can be generalised.
Take a Number N and the Number obtained by inverting the form
and the residue of N: The residue of N is promoted into the form,
whilst all the terms of its form are demoted into the residue. This
new Number operates, in the same ordinal-matter, a cut inverse or
symmetrical to that which defines N. We will call this Number the
symmetric counterpart of N (indicating a symmetry whose centre, as
we shall see, is Zero). We will denote by -N, and read as 'minus N',
the symmetric counterpart of N.
A Number and its symmetric counterpart can be presented as
follows (using the diagrams introduced in 12.3):
It is clear in the diagram that N is positive (0 is in its form) and
that -N, its symmetric counterpart, is negative. Evidently, this will
always be the case. Conversely, when N is negative (0 is in its residue),
-N is positive (0 is in its form).
If we take the symmetric counterpart -N of N, then the symmetric
counterpart -(-N) of -N, we arrive back at N: we have changed the
form into the residue, and then the residue into the form. It is that
old law learnt in the schoolroom, which spontaneously opposes itself
both to Hegel and to intuitionism: two negativising operations take
us back to the initial affirmation. However, one must still take care,
as always, to note that -(-N) is not necessarily a positive Number.
If the starting Number N is negative, its symmetric counterpart is
positive, and the symmetric counterpart of its symmetric counterpart
- which is itself - is once again negative. The sign '-' is not a sign of
negation but one of symmetry. Which confirms for us that the nega-
tive (unlike the symmetrical) is not an operational dimension. It is a
structural predicate of the being of Number.
16.9. A few examples.
What is the symmetric counterpart of the positive Number (O>,(O))?
It is the Number (0),(0> - (0))), whose form is all of 0> except for O.
It is obviously negative.
What is the symmetric counterpart of the negative Number (2,(1)),
whose form is the singleton of 1? It is the positive Number (2,(0)),
whose form is the singleton of (0). In fact, the only elements of the
ordinal 2 are 0 and 1. In the former case, 0 constitutes the residue,
in the latter, the form.
Take a positive Number N and its symmetric counterpart -N. To
every Number situated 'between' Zero and N we can make corre-
spond a Number situated 'between' -N and Zero: we just take its
symmetric counterpart. In fact, it is clear that, where it is the case
that Zero < N, < N, it is also the case that -N < -N, < Zero. This
can be verified by examining all possible cases of inequality between
N, and N (see 13.13), remembering that -N swaps the form and
residue of N.
There are thus 'as many' Numbers between -N and Zero as there
are between Zero and N, because the function ((NIl = -N, is a biuni-
vocal correspondence between the two 'slices' of Numbers. But take
care! The correspondence is not between two sets. The interval
between Zero and N is not a consistent totality any more than the
entire domain of Numbers is. This can easily be proved: taking, for
example, the Number (2,(0)), we know that all Numbers of the type
(W,(O)), where W is any ordinal whatsoever larger than 2, are smaller
than (2,(0)). It is the law that we discovered in 13.16: if the form
stays the same and the matter is increased, the Number gets smaller.
Meanwhile, all Numbers (W,(O)) are positive, since 0 is in their form.
So there are 'as many' of these positive Numbers - that is, those situ-
ated between Zero and (2,(0)) - as there are ordinals larger than 2.
But we know for sure that 'all ordinals larger than 2' is an inconsis-
tent multiplicity.
Keeping this in mind, we can allow ourselves to visualise symmetry
in the following way, the axis being that of Numbers taken according
to their order:
• • • •
This justifies our speaking of a symmetry whose centre is Zero.
16.10. The ordinals
We announced a long time ago (see for example 8.8) that the ordinals,
which constitute the stuff of the being of Numbers, can also them-
selves be represented as Numbers. What do the Numbers that
represent ordinals look like?
Let's consider the Number (W,W), whose matter is the ordinal W
and whose form retains all of this matter. In other words, this is a
case of a maximal numerical section, or of exhibiting - as certain
contemporary artists have done - the raw material alone as the
'work'. The most interesting thing is to compare the Number (W,W)
with the Number (W,O), whose form is void. In both cases, we feel
that the act is somehow null. But the two nullities are distinct. The
Number (W,W) treats the whole of the matter as a form, whereas the
Number (W,O) does not inscribe any form in the matter. The immedi-
ate result is that (W,W), for any W other than 0, is a positive Number,
whereas (W,O) is a negative Number (remembering that 0 is not an
element of 0, and that therefore 0 is not in the form of (W,O)). We
discern a certain positivity in the first gesture which designates the
matter as form, whereas the second, overwhelmed by the matter, is
unable to designate anything whatsoever.
But if (W,W) is treated as a positive production, the assumption
of a matter as form, it remains nevertheless a fact that this production
repeats the ordinal-matter. This redoubling of the ordinal (as matter,
then as form) legitimates our treating Numbers of the form (W,W)
as the Numerical representatives of the ordinals.
We will therefore say the following: An ordinal W is presented as
Number in the form (W, W); that is, the Number whose matter is W
and whose form is W. This presentation is the ordinal 'itself', but
thought as Number.
16.11. To be sure that this is the ordinal 'itself', we must explicitly
prove that the order of Numbers respects the order of the ordinals,
which is belonging. In other words, that if it is the case - ordinals
being thought in their own domain - that WI E W 2, then it is also
the case - ordinals being thought as Numbers - that (W"Wd <
This is obviously the case. Because the discriminant of (W"Wd
and (W
) is necessarily the smallest ordinal to belong to WI and
not to W
, or to belong to W
and not to WI' If WI E W
, this smallest
ordinal is precisely W" which belongs to W 2 but cannot belong to
itself. Now WI is outside the matter of (W"Wd, and it is in the form
of (W
). So it is indeed the case that (W"Wd < (W
Thus the order of the ordinals thought as Numbers, in the formal
redoubling of their material being, is the same as the order of ordinals
thought in their being, as transitive sets all of whose elements are
transitive. The Numerical representation of the ordinals is structur-
ally isomorphic to the ordinals. This being so, there is no reason why
we should not consider that the ordinals 'themselves' are inscribed,
identically represented, in the order of Numbers.
16.12. From the fact that an ordinal is a Number of the form (W,W),
three consequences can be drawn:
1 Every sub-Number of an ordinal is an ordinal. For, if (W,W) is
an ordinal, a sub-Number is of the form (w"wd, where WI E W.
It is therefore the ordinal WI'
2 All these sub-Numbers will manifestly be ordinals smaller than
the initial ordinal. It follows that they are all in the low set of
the initial ordinal, and that the high set, generally composed of
sub-Numbers larger than the Number, is empty here. This is a
characteristic property of ordinals thought as Numbers. Gener-
ally speaking, a sub-Number of the high set is a sub-Number
Nlwl such that WI is in the residue of N. But, in the case of
an ordinal (and this could be a definition of the ordinals), the
residue is empty. The high set of an ordinal is therefore also
empty; and, conversely, if the high set of a Number is empty,
then its residue is empty: its form coincides with its matter; it
is an ordinal. The canonical presentation of an ordinal will
therefore be of the form Lo(W)/O. But what is more, as the
low set has for its elements all ordinals smaller than W, it is,
as a set, identical to W (every ordinal is the set of all the
ordinals smaller than it, 11.2). Finally, the canonical representa-
tion - most distinctive - of an ordinal W thought as Number
is simply W/O.
3 The symmetric counterpart of an ordinal (W,W) is obtained by
swapping the residue and the form. Now, the residue is empty.
So it is the void that will be substituted for the 'total' form
that is W: the symmetric counterpart of (W,W) is the Number
(W,O). Thought as Number, an ordinal Wallows of a symmetric
counterpart, so we can freely speak of the Number -W.
It is clear that every ordinal apart from ° is a positive Number, since
its form, W, contains ° as an element. The symmetric counterpart
of every ordinal other than the void is therefore a negative Number,
as can be seen directly in writing (W,O). It will be found, moreover,
that all the properties of an ordinal Ware inverted by the passage
to -W. So that now every sub-Number of -W is the symmetric
counterpart -WI of an ordinal WI smaller than W; and it is the low
set of -W that is void, since - the form of -W being void - every
sub-Number of -W is larger than it; and, finally, the high set of -W
is identical to -W, with the result that the canonical representation
is: Ol-W.
We are thereby assured that ordinals are Numbers.l But what is
more, grasped in terms of Numericality, the ordinals are symmetricis-
able: we have opened up on the other side of Zero (which is the
ordinal 0, thought as Number) an immense space where will be
inscribed those formerly unthinkable entities: natural multiplicities
submitted to the negative. Numericality is capable of symmetricising
16.13. Positive and negative whole numbers
The natural whole numbers, thought in their being, are none other
than the finite ordinals, which is to say the elements of 0>, the first
limit ordinal. In fact we have already given their definition and dis-
cussed their operational dimensions in chapter 11.
This preceding work already settles the question, then: thought as
Numbers, natural whole numbers are of the type (n,n), where n is a
finite ordinal. Evidently, they are all positive. The order of natural
whole numbers qua Numbers coincides with the order of natural
whole numbers that we already know, the order according to which
every schoolboy says that n is larger than p. For we know that, if
pen - which is the ontological version of traditional order - then
(p,p) < (n,n) in the order of Numbers. We therefore have the right to
write the Number (W,W) as W, to indicate that an ordinal 'itself' is
being inscribed in the domain of Numbers. We therefore write a
natural whole number, thought as Number, as n.
The sub-Numbers of a natural whole Number are the finite ordi-
nals smaller than it, therefore the natural whole numbers smaller than
it. If n is this Number, these will be natural whole Numbers
(0,0),(1,1), ... , ((n - l),(n - 1)), which we could also write as
0,1, ... , (n - 1). Taken together, they form the low set of n. The high
set of n is empty, and the canonical representation of a whole n,
thought as Number, is (0,1, ... ,(n - 1))/0. Since n's elements are
precisely 0,1, ... ,(n - 1), the low set whose elements they are can be
written as n/O. (NB This is not circular, because, considered as a set,
n does not contain itself as an element).
The symmetric counterpart of a natural whole number is a Number
of the form (n,O), where n is a finite ordinal. We write it -n, we say
'minus n'. We posit that a Number is a whole negative Number if it
is the symmetric counterpart of a natural whole Number, that is, one
which takes the form (n,O). The sub-Numbers of a negative whole
number -n are all the whole numbers -p, where pen. Taken
together, they form the high set of -n, whose low set is empty. The
canonical representation of a negative whole Number is therefore
ultimately written as Of-no
In order to confirm the complete identity of the traditional positive
and negative whole numbers and of the positive and negative whole
Numbers, it must obviously be the case that operations on these
Numbers coincide, as order did, with operations on numbers. If for
example we define an addition NI + N
on Numbers, then the result
of this operation in the specific case of whole Numbers, hence in the
case of an addition of the type m + n, should be 'the same Number'
as the whole number which, in the calculations of our schooldays,
corresponded to the addition of these two whole numbers. These
operational verifications will be carried out in chapter 18.
So far as the inscription within the Numbers of natural whole
numbers thought in their being is concerned, our task is complete.
16.14. Dyadic positive rational numbers
We have already spoken of rational numbers in relation to Dedekind
cuts (compare 15.5): a positive (or null) rational number is a fraction
or relation % of two natural whole numbers, which is to say a pair
(p,q) of whole numbers. The first is called the numerator, the second
the denominator. The numerator can be null (identical to the empty
set), but it is prohibited for the denominator to be 0 (we know that
the relation tr is 'undetermined').
We have no desire here to enter into a rigorous introduction to
these traditional numbers (in fact, here we must consider fractions as
irreducible, impossible to simplify). The intuitive idea of the fraction
will suffice for us.
It is evident that the natural whole numbers are a subset of rational
positive or null numbers; we just need to take a rational in the form
t to obtain n. In other words: a whole number is a rational of the
type (n,1).
The classical order of the rationals has the fundamental property
of being a dense order. In other words (see 15.5), given two rationals
* and :: such that * < ~ ,however 'near' these two numbers might
be, there always exists a third (and, from there, an infinity of them)
which comes between the two initial numbers: there is a :' such that
~ ~ ~ ,
q, q, qz'
A dyadic rational number is a number of the form J. whose
denominator is a power of 2. Or, in our paired version, a rational
number (P,2°).
Dyadic rational numbers themselves form a dense subset of the
rationals: if rl and r2 are rationals such that rl < r2, a dyadic rational
can always be intercalated between them.
The important thing for us is that every sequence of augmenting
rationals rl < r2 < ... < ro < ... can be 'replaced' by a sequence of
dyadic rationals d
< ... < do < ... : take the dyadic rationals situated
'between' rl and r2, then r2 and r3, etc. We can also say that the dyadic
rationals form a 'basis' for all the rationals. More specifically, a non-
dyadic rational number can be 'approached' as closely as you like
from a dyadic one, because you can always lodge a dyadic in-between
rand r + r', however small r' might be.
16.15. We have, then, the following statement, perhaps the most
important in the process of the representation of (traditional) numbers
as (ontological) Numbers:
Every dyadic rational number can be represented as a Number of
finite matter, and every Number of finite matter represents a dyadic
rational number.
16.16. How, in general, is a Number of finite matter presented? In
the form (n,(pt.P2,'" P,)), where the whole numbers PI. P2, etc.
which make up its form are whole numbers smaller than n, the matter
of the Number. Since we are keeping to rational positive numbers,
we will consider here only Numbers of positive finite matter, that is,
Numbers which have ° in their form.
The subtle idea that guides the 'projection' of these Numbers of
finite matter into the dyadic positive or null rationals is the following.
Let n be the matter of the Number. We take all the elements of this
matter in order, from ° to n - 1, which is the largest whole number
contained in n. In so far as we stay in the location of the first element,
° -which is the form, since the Number is positive - we attribute
the value 1 to the whole number in question. Say that we come across
the first element of n - say p - not to have the same location as 0,
in other words the smallest whole number p in the matter of the
Number to be in the residue. We attribute to this whole number the
value -±. After this, we generally attribute to whole numbers q
which follow the value zf.;r if they are in the form, the value -zf.;r
if they are in the residue.
Finally, the value of the last term beyond p (p being still the first
term which does not have the same location as 0, if it exists), the
value attributed to n - 1, then, will be *, with or without the sign
depending on whether n - 1 is in the residue or in the form.
Or, once again: a belonging to the residue will always be affected
by the sign -. In traversing in order all the elements q of n, in so far as
one remains within the form, which is the location of 0, each element
is counted for 1, for a whole value. As soon as the location changes,
we count the elements for a dyadic rational of the form zf.;r, where
p is the first for which the location changes, from now on adding the
sign - whenever this location is the residue.
Finally, we associate with the initial Number of finite matter the
rational number obtained from the sum (in the usual sense) of all
the values thus attributed to the elements of n. This rational number
is dyadic, since all the denominators in question are dyadic, and
since - as every schoolchild knows - to add fractions, one takes as
denominator the smallest common multiple of the denominators.
Now, the smallest common multiple of powers of two is a power
of two.
16.17. Let's give an example of the procedure. Take the Number
(5,(0,1,3)), whose (finite) matter is the ordinal 5 and whose form
contains 0, 1 and 3. The residue is thus composed of 2 and of 4.
Since ° is in the form, we give it the value 1.
Since 1 is also in the form, we give it the value 1.
The location changes with 2, which is in the residue. We give it
the value -to
·· h f .. h I 1 1
IS In t e orm; we gIve It t e va ue 2'-2,1 = 2' •
4 is in the residue, we give it the value - /2.1 =-fr.
So, in the end, the rational number corresponding to the Number
(5,(0,1,3)) will be obtained from the sum:
1 1 1 13
2 22 2
We can see very well that this is indeed a dyadic rational.
16.18. In order better to exhibit the construction of this correspon-
dence, which bears witness to an isomorphy, an identity of being,
between positive Numbers of finite matter and positive dyadic ratio-
nals, we will formalise things a little. We will then see clearly that we
are dealing with an inductive definition, a definition by recurrence.
Take a positive Number of finite matter. We will define by recur-
rence the following function (, defined on the elements of the matter
n of the Number:
RULE 2: ((p + 1) = 1, if ((p) = 1 for all whole numbers up to and
including p, and if p + 1 is in the form of the Number.
RULE 3: ((p + 1) = -t if all the whole numbers up to and includ-
ing p are in the form and p + 1 is in the residue.
RULE 4: ((p + 1) = 2!.1 if the value of p is iq or - 21q and p + 1
is in the form.
RULE 5: ((p + 1) = ~ if the value of p is f.- or -f.- and p +
1 is in the residue.
These rules will allow us to calculate the rational value of ( for all
the elements of n, the matter of the initial Number. Using Ra(N) to
denote the dyadic rational that corresponds to N, we then posit
Ra(N) = ((0) + ((1) + ... + ((n - 1)
The sign + indicates here the algebraic sum in the normal sense.
It is clear that Ra(N) is a dyadic rational.
16.19. Let's proceed with the calculation of another example, the
Number (4,(0,1,3)), which is, of course, a positive Number of finite
((0) = 1 (by rule 1).
((1) = 1 (by rule 1).
((2) = -.!. (by rule 3; 2 is in the residue).
1 1
+'(3) = - - - (rule 4,' 3 is in the form).
(' 21+1 - 22
Ra((4,(0,1,3))) = ((0) + ((1) + ((2) + ((3).
1 1
Ra((4,(0,1,3))) = 1 + 1-"2+2:2'
Ra(( 4,(0,1,3))) ~ which is a dyadic rational, as we said it
would be. 2
16.20. Whole ordinal part of a Number
It might appear strange peremptorily to change the procedure when
we get to the first whole p that doesn't have the same location as
o in the Number of finite matter under consideration. Gonshor
realises this: 'The whole idea of a shift from ordinary counting to
a binary decimal computation at the first change in sign may seem
unnatural at first. However, such phenomena seem inevitable in a
sufficiently rich system.,2 This explanation of Gonshor's - more of
an apology, really - is a little quick.
To find the true underlying concept, we should ask what is actually
represented by the p first consecutive ordinals of a Number N of
finite matter which have the same location as the initial term O.
Assume once more the positive case (0 located in the form). If we
partition N at point p (the first ordinal, in the order of ordinals, to
change location), we obtain the sub-Number NIp all of whose ele-
ments have the same location as o. It is clear that, since this location
is the form, NIp is the whole positive Number p, that is, the Number
whose matter is p and whose form is made up of the elements of p.
The function f will attribute the value 1 to all these elements, and
the sum of the values 1 + 1 + ... will give the 'classic' whole number
p. Which, we can add immediately, is an algebraic sum of dyadic
rationals of the type i
• or - 21q , where q is no more than 1. It follows
that Ra(N) will be the sum of the whole p and a negative dyadic
fraction between -1 and 0 (at least, unless it happens to be a whole
number). Finally, p is a type of whole part of the positive rational
Ra(N), that is, the natural whole number closest to Ra(N) 'from
above': (p - 1) < Ra(N) < p.
From the point of view of Number, in fact, p is the largest sub-
Number of N to he an ordinal, since 'being an ordinal' means pre-
cisely being a Number all of whose matter is in its form. That the
location changes at point p (p is in the residue) means precisely that
NIp + 1 is no longer an ordinal either, since p, an element of its matter,
is in the residue. It is therefore even more fitting to say that p is a
'whole part' of N. By which we mean: the largest whole number p
belonging to the matter of N and such that the sub-Number NIp is
the ordinal p. Or even more simply: the whole part of N is the largest
ordinal to be a sub-Number of N.
Now the procedure becomes clearer: it works firstly by making
correspond, via f, the elements of the whole part of N and the
whole part 'from above' of the dyadic rational Ra(N). The 1 values
are used to do this. And then it is a question of calculating the
remainder, which is less than 0, but more than -1, and to do this
we use dyadic fractions of the type j; or -j;, q indicating the
rank of the ordinal in question beyond the whole part p. There
is no 'unnatural' mystery in all of this, but rather a profound
16.21. We can generalise these remarks. Given a positive Number N
of matter W, we will call whole ordinal part of N the largest ordinal
WI E W such that the sub-Number Nlwi is the ordinal WI.
The attentive reader will balk at this: how can we speak so freely
of the 'largest ordinal' to satisfy a property? Doesn't the existence of
limit ordinals militate against any such claim? Where ordinals are
concerned, only minimality is at work.
The remark is well taken. We will have to reformulate our defini-
tion, then, and posit the following: The whole ordinal part of a posi-
tive Number is the smallest ordinal located in the residue. Since the
Number is positive, 0 is located in the form. The smallest ordinal
located in the residue is thus indeed the first ordinal, in the ascending
order of the ordinals of the matter of N, all of whose elements are
in the form, although it itself is in the residue. These elements con-
stitute the whole ordinal part of N. Here is a case where 'the largest'
translates as 'the smallest'.
If WI is the whole ordinal part of a positive Number N then, just
as in the above case, N < w" since the sub-Number WI is, considered
as an element of the matter of N, in the residue of N, whereas it is
outside its own matter.
It can also be said that the whole ordinal part of a positive Number
is in the high set of that Number.
If WI is a successor ordinal, once again we find the 'framing') of
the endpoint. Let W2 be the predecessor of WI; this gives WI = S(W2)'
Since WI is the smallest ordinal to be in the residue, its predecessor
W2 must be in the form. Of course, since all the elements of W2 are
elements of WI (transitivity of ordinals), and all the elements of WI
are in the form, all the elements of W2 are too; so N1wz is the ordinal
W2. And, given that this ordinal is outside its own matter and in the
form of N, then W2 < N, and so finally W2 < N < S(W2) = WI' This is
the interval we are looking for.
If, on the other hand, WI is a limit ordinal, it will certainly always
be the case that N < WI, but we would search in vain for the largest
ordinal smaller than N, because on the other side of WI there is no
'predecessor'. N would then have a singular position: smaller than a
limit ordinal, it would be larger than all the ordinals smaller than
this limit ordinal. It would come to insert itself in that space we
thought was 'filled in' by the ordinals that precede the limit, the space
'between' a limit ordinal and the infinity of successor ordinals of
which it is the limit.
16.22. Let's give an example. Take the Number N = (S(O)),S(O)) - (0))),
whose matter is the successor of 0) and whose form is all of that
matter except for 0) itself, which is the only element of the residue.
The limit ordinal 0), being the first ordinal in the matter of N to be
in its residue, is the whole ordinal part of N. It is indeed the case that
N < co, since their discriminant is co, which is in the residue of Nand
outside the matter of co. What is more, for every element of co - that
is, for every natural whole number n - it is the case that n < N, since
n is outside the matter of n and in the form of N. The Number N is
thus at once smaller than the first limit ordinal co and larger than all
the natural whole numbers n of which co is the limit! This shows to
what extent the domain of Numbers saturates that of the ordinals,
which it contains: there are 'many more' Numbers than there are
We can also say that N is 'infinitely near' to co, far nearer than even
the most immense of the whole numbers could be. This notion of
'infinite proximity' is of a prodigious philosophical interest. It opens
up new spaces for exploration in the endless kingdom of Number. We
shall undertake these explorations a little later.
16.23. Sequence and end of the dyadic rationals
We have at our disposal a function Ra(N) which makes a dyadic
rational correspond to every Number of finite matter. The whole
numbers are included in this correspondence, because the positive
whole number n thought of as Number will correspond, through the
function Ra, to the sum 1 + 1 + ... + 1 n times - that is exactly the
Number n, since, if a Number is a natural whole number, then all of
its sub-Numbers are in its form. It would be better to say that the
function Ra associates a dyadic rational with every Number of finite
matter - even if this Number is whole.
To complete the work, and to conclude that the dyadic rationals
'themselves' are represented in Numbers, we must:
• confirm that the order of Numbers of finite matter is isomorphic
with the customary order of corresponding dyadic rationals, so
that, if Nl < N2 in the order of Numbers, then Ra(Nd < Ra(N
in the normal order of rationals; this amusing mathematical exer-
cise is sketched nicely in the note;4
• prove that all the dyadic rationals are obtained through the func-
tion Ra applied to Numbers of finite matter; this comes down to
proving that every positive dyadic rational can be put in the form
of the algebraic sum of a certain whole number (its whole part
'from above') and dyadic rationals of the form +.- or -+.- ; because,
once this is done, one can reassemble the Number N, whose value
for Ra is the rational thus dismembered;5
• prove that the operational dimensions of the rationals - addi-
tion, multiplication, division, in brief, everything that gives them
the algebraic structure of a field, are isomorphic to the same
operations defined for Numbers and applied to Numbers of
finite matter; this relates to the examinations made in chapter
18, with one obvious exception: in order to have negative dyadic
rationals, the procedure of symmetricisation would be used,
which defines the general manner of passage to the negative:
inversion of swapping form and residue. Of course, we will still
be dealing with a Number of finite matter (but this time with
o in the residue).
As far as the ontological side of things is concerned, we have
attained our goal. A dyadic rational, thought in its being, inscribed
as Number, has a very simple intrinsic definition: its matter is finite.
As far as being is concerned, that, this clarifies however dense the
rationals might be, even to the point of an infinite swarming between
two consecutive whole numbers, they nevertheless belong to the
finite. The numerical ontology of the infinite begins with real
16.24. Real numbers
We know that real numbers provide the model for the geometrical
'continuum': their figure is that of the points of a line. It is the
real numbers that have subtended the entire edifice of analysis,
chef-d'oeuvre and keystone of modern mathematical thought,
since Newton and Leibniz.
For a long time, the continuum and the functions corresponding
to it were thought either in terms of geometrical constructions (Greek
and pre-classical age), or in a primitive and pragmatic fashion (eigh-
teenth and nineteenth centuries). The emergence of a rigorous concept
of reals as entities with which one can calculate took place slowly
during the course of the nineteenth century, beginning with Cauchy,
and with Dedekind representing a decisive step.
Because it is the closest to that which governs the definition of
reals in the field of Numbers, we will recall briefly the construction
of real numbers by means of 'cuts', as invented by Dedekind.
16.25. We will begin with dyadic rationals, which we can use here
in place of rationals as such, in view of the remark made in 16.14.
Take two sets of dyadic rationals B and A such that every rational in
B is smaller than every rational in A. We can say both that B has no
internal maximum (for every dyadic rational r1 in the set there is an
r2 in the set such that rl < r2); and that A has no internal minimum.
Suppose now that the following relation holds between B and A: there
always exists a dyadic in B that is 'as close' as one likes to a dyadic
of A. In other words, if r2 is a dyadic in A and r a dyadic as small as
one likes, there will always exist a rl in B such that the difference
between r2 and rl is less than r.
The situation can be visualised as below, by representing the dyadic
rationals as points on a line:
• • • •
reB r'eA
We can see clearly that B 'rises' without ever entering into A, that
A 'descends' without entering into B, and that therefore the two sets
are as close to each other as can be, without ever 'touching'.
Thus did Dedekind define a real number as the point situated
exactly 'between' B and A; that is, the element, created in this process,
which is simultaneously larger than any element of B and smaller
than any element of A. We can identify this element as the point of
the cut of Band A.
It is characteristic of this method that it treats the cut not as a state
of things in a pre-given universe (which is how we treated it for
Numbers, see 15.6), but as a procedure, defining a mathematical
entity that does not pre-exist this procedure. To begin with, there are
only rationals. And, if the cut is not a rational (it could be, if the
upper limit of B and the lower limit of A coincided), then it consti-
tutes in itself the name, or form of presentation, of a 'being' which
inexists in the field of rationals. Therefore the reals are operational
productions here; they sign, coming forth from non-being, the fictive
point where B and A are touched by the interposition between them
of this fiction. Into that place, where there was nothing but the min-
iscule void that separates two sets as close as can be, comes the real,
which stops up this void by realising a cut as number.
16.26. Fictions have no place in the ontological conception of
Number. If the classic real numbers, those which realise cuts in the
dyadic rationals, inscribe themselves in the domain of Numbers, it is
because they exist and are distinguished by some property. They
cannot irrupt from inexistence, in the form of mere names of a
lacuna. According to an ontological conception of Numbers, every
Number is, none results or is resolved in the name of an operation.
We do battle here against a dominant nominalism, and we do so in
the field of number, so commonly taken for an operational fiction.
16.27. In fact, our definition of real numbers as Numbers is quite
A Number is a real number if it is either of finite matter or of
matter 0), and if its form and its residue are infinite.
In what follows we will substantiate this definition, which repre-
sents real numbers 'themselves' in the domain of Numbers.
16.28. The 'projection' of this definition into the concept of reals as
cuts is basically very simple.
If a Number is of finite matter, then it is, as we have seen, a dyadic
If a Number is of matter 0), then all of its sub-Numbers are of
lesser matter than 0), and therefore of finite matter, since 0) is the
smallest infinite ordinal. So all of its sub-Numbers are dyadic ratio-
nals. More specifically, its low set and its high set are sets of dyadic
rationals. And, since every Number is the cut of its low set and its
high set, a Number of matter 0) can be represented as the cut of two
sets of dyadic rationals. Or, once again, a real number thought as
Number is a Number whose canonical presentation Lo(N)/Hi(N) is
made solely from dyadic rationals.
Finally, if a Number of matter 0) has an infinite form and an
infinite residue, we avoid its low set and high set having internal
maxima. Because, if the form of N is finite, since it is composed
of whole numbers (the matter being 0)), it admits of a largest element,
say the whole number p. The cut of N at point p defines the sub-
Number NIp, which is obviously the largest sub-Number whose
discriminant with N is in the form of N, and therefore the largest
sub-Number in the low set of N. And, if the residue is finite, there
exists a number p such that NIp is the smallest element of the high
set of N. A cOlltrario, if both the form and residue of N are infinite
- are sequences of whole numbers without internal maxima - then
the low set does not have a maximum term, nor the high set a
minimum term.
We thus find ourselves precisely in the conditions of the Dedekind
cut: disjoint ascending and descending sets of dyadic rationals with
no maximum or minimum. Except that what we characterise as
'reals' are particular, already existing Numbers, whereas Dedekind
installs them as a fiction at the void point of a cut. For us, a real
will be that unique Number of minimum matter situated exactly
between two sets of dyadic rationals which can be shown to be its
low set and its high set, and therefore to be sets of sub-Numbers.
It is particularly reassuring to remark that, in the definition of
reals as Numbers, everything remains immanent. Dedekind cuts des-
ignate the fiction of a number external to two sequences of rationals,
as the point of contact of these sequences. Whereas, on the other
hand, the sets of dyadic rationals that we use are composed of sub-
Numbers of a real Number. This immanentisation of procedures is
typical of the ontological approach, that approach which captures
the being of Number. To see if a Number that is not a dyadic rational
is a real number, it suffices to examine it according to its three
- its matter must be (0;
- its form must be infinite;
- its residue must be infinite.
This alone allows us to conclude. Then we can state that the
Number is the cut of two sets of dyadic rationals, and that therefore
it is indeed a real number (in the classic sense). But, all the same, we
have remained within Number, since dyadic rationals are sub-Numbers
of a Number.
The immanence of the thinking of being has not faltered for a
moment in this approach to the traditional real numbers grasped in
the space of Numbers. The characterisation of a type of pure multiple
has been substituted for operational fictions. And real numbers are
no more mysterious here than whole numbers or rationals. Their sole
peculiarity is that they mark the moment where our passage through
Numbers prompts us to envisage infinite matters. From this point of
view, the ontological singularity of the reals in relation to the wholes
and the rationals can be summed up in one word: infinity. This alone
clarifies, irrespective of all complexities of construction, with an eye
only to that in which the numerical section operates, the fact that
real Numbers are exemplarily modern.
16.29. We now find ourselves in possession of a concept of Number
that subsumes as particular species the natural whole numbers, the
whole positives and negatives, the rationals, the reals, and the ordi-
We have overcome the modern resistance to a unification of
the concept of Number (see 1.8). But, in the process, we have already
seen that this concept also subsumes other Numbers, that the histori-
cal deduction from the domain of Numericality is very much limited.
Rationals and reals cover the totality of Numbers of finite matter and
only some Numbers of matter (0. It is as if our thinking has so far
only brought to light a minute initial segment of that which being
proffers in terms of possible numerical access to pure multiplicities.
The future of the thinking of Number is limitless.
16.30. Infinitesimals
We remarked in 16.22 that it would be possible to find a Number N
at once smaller than 0) and larger than all the finite ordinals whose
limit is 0). This Number could perhaps be said to be 'infinitely close'
to 0), and it puts us on the way to a concept of infinitesimal Number.
The idea of infinitely small number, freely employed by seventeenth-
and eighteenth-century mathematicians, was dismissed in the nine-
teenth century for its obvious inconsistencies. It was replaced by the
concepts of the limit (Cauchy) and of the cut (Dedekind). It reap-
peared around thirty-five years ago, in the singularly artificial, but
consistent, context of the pure logic of models: Robinson's non-
standard analysis.
In the domain of Numbers, 'infinitely small
numbers' or infinitesimals abound in the most natural fashion. It is
by means of them that we will complete this diminutive journey
through the enchanted kingdom of Numbers.
16.31. Consider the Number i = (0),(0)), whose matter is 0) and
whose form, the singleton of the void, has the void as its only element.
It is a positive Number, since 0 is in its form.
Now this positive Number, even if its matter is the same as that
of real Numbers, is smaller than every positive real Number.
In fact, if a real Number is positive, 0 is in its form, as is the case
for i: 0 does not discriminate between i and a positive real Number.
All the whole Numbers other than 0 being in the residue of i, the
discriminant of i and a real Number R wiII be the first whole Number
apart from 0 to figure in the form of R. Such a Number necessarily
exists, since the definition of the reals dictates that the form of R
should be infinite. And, since this discriminant is in the residue of i,
i is smaller than R. Therefore there exists a Number i such that
0< i < R for every real Number. This i is situated 'between' Zero and
all real numbers thought as Numbers. We will say that it is infinitesi-
mal for the reals.
16.32. Generalising this definition: We say that a set of positive
Numbers, all of the same matter, tends rationally towards Zero if,
for every dyadic positive rational r, as close to Zero as you like, there
exists a Number N, of the set situated between Zero and r. In other
words: for every dyadic rational r, there exists N, belonging to the
set such that 0 < N, < r. Note that the classic notion of 'tending
towards' is here relativised to dyadic rationals. In the limitless domain
of Numbers, we must indicate which scale of measurement is being
employed, because, as we will see, it is always possible to find a still
finer scale.
It is obvious that the set of real positive Numbers tends rationally
towards Zero. Other sets of Numbers can be found which tend ratio-
nally towards Zero, for example positive Numbers of the type
(S(co),(O, ... )), whose matter is S(co) and whose form contains at
least O.
We can say, then, that:
A Number is infinitesimal for a set of Numbers that tends rationally
towards Zero if it is:
- of the same matter as the Numbers of the set;
- positive;
- smaller than all the Numbers in the set.
So it is that the Number (co,(O)) is infinitesimal for the set of real
Numbers. On the other hand, there is no infinitesimal for the set
of Numbers (S(co),(O, ... )), precisely because this set contains the
very Number (S(co),(O)) that is the smallest positive Number of
matter S(co).
The limiting of the concept of infinitesimal to Numbers of the same
matter as the Numbers of the set that tends towards Zero is necessary
because, if this restriction were not in place, there would be as many
in{initesimals as we wished. It would suffice to augment the matter:
the Number (S(co),(O)) is positive, and it is certainly smaller than every
positive Number whose matter is co. In particular, it is smaller than
the infinitesimal i = (co,(O)), because the discriminant is co, which is
outside the matter of i and in the residue of (S(co),(O)). We see to what
extent our concept of the infinitesimal is relative: the density of order
over Numbers means that, however 'relatively' small a positive
Number might be, there still exists an inconsistent multiplicity of
Numbers situated between it and Zero.
We can, if we wish, retain the classic definition: every positive
Number smaller than every positive real is infinitesimal. But then we
will see the infinitesimals grow and swarm uncontrollably. The
'shores' of Zero contain 'as many' Numbers as the entire domain of
Numbers. Because, at the point where multiple-being as such incon-
sists, the notion of 'as many' loses all meaning.
16.33. Cuts of cuts
Take the Number C = (co,(0,1)), whose matter is co and whose form
is limited to the wholes 0 and 1. This Number is not real, since its
form is finite. It is positive, since 0 is in its form. How can it be
situated amongst the reals, to which its matter belongs?
A positive real which does not have 1 in its form is certainly
smaller than C: the discriminant is 1, which is in the residue of such
a real and in the form of C.
A positive real which does have 1 in its form is certainly larger
than C. For all whole numbers larger than 1 are in the residue of C,
whereas some of them are certainly in the form of a real, since this
form is infinite. The discriminant will be the smallest whole larger
than 1 to be in the form of the real, and, since it is in C's residue, C
will be smaller.
C therefore is situated precisely between the reals which have 1 in
their residue and the reals which have 1 in their form. Now these two
classes operate a partition into two of the positive reals, a partition
which is ordered (all the positive reals which have 1 in their residue
are smaller than all the positive reals which have 1 in their form). We
can, then, perfectly lodge a Number 'between' two disjoint classes of
reals, in the caesura of a partition of reals. And, since the reals are
themselves cuts of rationals, the Number C will be a cut of cuts.
Generally speaking, given an organised partition into two of a set
of Numbers 'of the same type', that is to say, defined by cuts or
canonical presentations having this or that property (as we saw in
defining the reals), we will call a 'cut of cuts' a Number of minimal
matter situated in the caesura of the partition, being larger than all
those in the lower segment and smaller than all those in the higher
segment. The Number (00,(0,1)) is a cut of cuts in the numerical type
'positive real Numbers'.
The existence of cuts of cuts attests once more to the infinite capac-
ity of Numbers - as coalescent as they might seem - for cutting at a
point the ultra-dense fabric of their consecution.
16.34. So many other Numbers to visit and to describe! But works
that take delight in this are beginning to appear. And the philosopher
is not defined by curiosity; the journey is not a disinterested one. The
philosopher must, before leaving the kingdom convinced that every
number thought of in its being is a Number, descend back down to
calculation. Or, rather, to the existence of calculation, because the
philosopher is not a calculator either. But these numbers, from which
our soul is knitted, the philosopher wishes to render over entirely,
even as regards the derivation of their operational mechanism, to the
immemorial and effectless transparency of Being.
Operational Dimensions
Natural Interlude
17.1. The domain of ordinals (and of cardinals) holds an extreme
charm for thought. A proof by affect - by affection, even - of what
I claim here, is that, on reflection, this charm is that of Nature itself:
an abundant diversity and, at the same time, a mute monotony.
Nothing is the same, everything goes to infinity, but one hears a fun-
damental note, a basso ostinato, signalling that these myriads of
multiplicities and forms, these complicated melodies, proliferate
the repose of the identical. If poets' metaphors take as their
reference the sky and the tree, the flower and the sea, the pond and
the bird, this is because they would speak this presence of the Same
that the unlimited appearances of nature veil and reveal. In the
same way, the ordinals, still singular in the infinity of their infinite
number, in the inconsistency of their All, also repeat the transitive
stability and the internal homogeneity of natural multiples, those
multiples that they allow to be thought in their pure being. It is hard
to tear oneself away from the intellectual beatitude brought on by
the contemplation of the ordinals, one by one and as a 'set'. I think
of the great Indian mathematician Ramanujan, I who held each whole
number to be a personal friend. He was invested by this poem of
Number, of which the Poem of nature is the symmetrical counterpart
within language. He did not like to construct proofs, but rather, as
a dreamer of the ordinal site, to draw in it with curves of recognition,
which his colleagues regarded with some surprise. Coming from afar,
in all senses of the word, he was not accustomed to our severe modern
distinctions. He saw numbers directly for what they are: natural
treasures, where being lavishes its multiple resource and its fastidious
identity in the same gesture in which, for the poet, it arranges the
'correspondences' of sensibility.
17.2. We have at our disposal a concept of Number, and we know
that this concept subsumes our traditional numbers. Wholes, ratio-
nals, reals, ordinals, thought in their multiple-being, are Numbers.
It must now be shown - a slightly less rewarding task - that this
concept subsumes our traditional numbers not only in their being,
but also in their operations. As far as we may be from that sensibility
that is ruled by counting, it must nevertheless be shown that it is
possible to count with Numbers, and that this counting coincides,
for the classical types of Numbers, with ordinary counting. We must
cover algebra, addition, multiplication, etc. If we did not, then who
would believe us when, speaking from the sole point of view of being,
we said that these Numbers are numbers?
17.3. What is meant by 'operation', or calculation, is the consider-
ation of 'objects' upon which one no longer operates one by one, but
at least two by two: the sum of x and y, the division of x by y, etc.
And, as the matter of Number is made of ordinals, it is to be expected
that we have to deal with, to think, pairs of ordinals. So we will be
happily detained for a few more moments in the enchanted domain
of natural multiples. This whole interlude is dedicated to some reflec-
tions and propositions about pairs of ordinals, ordinals taken two by
two. And, as we shall see, these couples are also totally natural: we
can connect them back to 'single' ordinals via a procedure which in
itself holds a great charm.
17.4. We will speak of ordered pairs of ordinals, which we denote
by (W"W
). 'Ordered' meaning that one takes into consideration the
order of the terms in the couple - we will thus speak of the first term,
W" and the second, W
- which wasn't the case in our concept of
the simple pair, denoted by (e"e2) (compare 7.7), which was a pure
'gathering together' of two terms regardless of their order. Or, in
other words: if W, and W
are different, then the ordered pair
) is not the same thing as the ordered pair (Wz,W,). In order
better to distinguish the simple pair from the ordered pair, we will
call the latter a couple.
We can also allow 'couples' of the type (W"W,). In such cases, W,
occupies both the first and the second place.
The concept of ordered pair, or couple, plays a decisive role in
mathematics: it underlies all thinking of relations and of functions.
It can be reduced to a figure of the pure multiple, testifying to the
fact that relations and functions do not depend on any sort of addi-
tional being apart from the multiple, that there is no ontological dis-
tinction between bound objects and the bond which binds them. But
we will employ the concept here in its naiVe sense.
17.5. We will call maximal ordinal of a couple (W
), and denote
by Max (W
»), either the larger of the two ordinals WI and W
if they are different, or, if the couple is of the type (W h WI) the single
ordinal WI that figures in it. You are reminded (see 8.10) that ordinals
are totally ordered by belonging: if WI and W 2 are different, then one
is necessarily smaller than the other (belongs to the other).
This most elementary notion of the maximal term of a couple will
playa crucial role in what follows. It is important to get a firm grasp
of it.
17.6. Take a couple of ordinals (W
), which we will denote by
and another couple (W
), which we will denote by C
• We
will define an order-relation between these couples in the following
way. We say that C
is smaller than C
and write C
, if one of
the three following conditions is satisfied:
1 The maximal ordinal of the couple C
is equal to the maximal
ordinal of the couple C
• In other words: if Max(Cd E Max(C
it is always the case that C
2 The maximal ordinal of couple C
is equal to the maximal ordinal
of the couple Cz, but the first term of the couple C
is smaller
than the first term of the couple C
• In other words, in a case
where Max(Cd = Max(C
), if WI E W
, then C
3 The maximal ordinal of the couple C
is equal to the maximal
ordinal of the couple C
, and the first term of the couple C
equal to the first term of the couple C
, but the second term
of C
is smaller than the second term of C
• In other words,
Max(Cd = Max(C
) and WI = W
, but W
• In this case,
Evidently, if none of these three conditions are satisfied, then the
couples C
and C
must be identical: they have the same first term
and the same second term. A contrario, if two couples of ordinals
are different, either one is smaller than the other, or the other is
smaller than it: the relation is total.
This order follows directly from employment of the operator
Max(C), or, if this yields only an identity, from the comparative
examination, in this order, firstly of the first ordinal of each couple,
then, if this examination too yields only an identity, of the second
ordinal of each couple. The maximum trumps the first term, and the
first term the second. 50me examples:
• (6,0) is smaller than (0,7), because its maximum is 6, which is
smaller than the maximum of the latter, which is 7;
• (0,00) is smaller than (1,00), because, their maxima being identical
(it is (0), the first term of the former, 0, is smaller than the first
term of the latter, 1;
• (5(00),00) is smaller than (5(00),5(00), because, their maxima being
equal (they are both 5(00), the successor of (0) and their first term
likewise (it is 5(00) in both cases), the second term of the first
couple, which is 00, is smaller than the second term of the other,
which is 5(00).
Note that the couple (0,0) is the smallest couple of all, since its
maximum, its first term and its second term are all equal to 0, which
is itself the smallest ordinal.
It is also clear that couples form an inconsistent multiple, since,
already, the ordinals themselves cannot form a set. In speaking of
'the' couples, but also of 'the' ordinals, or 'the' Numbers, we must
always remember that we cannot attribute any property to whatever
this 'the' designates: there is no question of a thinkable, or present-
able, totality. In particular, if there exists a minimal couple for the
order that we are going to define (it is the pair (0,0»), there certainly
is not a maximum couple for this order.
17.7. I leave the reader the task of showing that the relation between
couples that we have just defined is a genuine order-relation (and
therefore, essentially, transitive: if C] < C
and C
< C), then
Far more interesting is the fact that it is a well-ordered relation. I
have given the definition of this in 6.4: given any set whatsoever of
terms well-ordered by a relation <, there exists one (and one only)
element of that set that is minimal for the order-relation, which is the
smallest element of that set.
Take any (non-empty) set E of couples of ordinals - that is, a set
all of whose elements are couples of ordinals. Consider all those
couple elements of E whose maximum is minimal for E. In other
words all the couples C E E such that Max(C) is the smallest ordinal
to figure in the elements of E as maximum of a couple. This is possible
by virtue of the principle of minimality that characterises the ordinals
(see 8.10). Given the property 'being a maximal ordinal in a couple
C which belongs to E', there exists a smallest ordinal to satisfy this
property. We thus obtain a subset E' of E, all of whose elements C
have the same minimal maximum. Note that, because of the first
of the conditions defining the order of couples, all the elements of
E' are smaller than all the elements that remain, that are in E-E'
(if any).
Now consider, in E', the set of couples whose first term is minimal
for E'. In other words, all the couples C = (Wt.Wz) such that WI is
the smallest to be found in all the first terms of the couples in E'.
This is possible for the same reason as before: it suffices to consider
the property 'being an ordinal that figures in a couple in E' as the
first term', and to take the minimal ordinal for this property. We will
thus obtain a set E" of couples having the same maximum (because
they are in E') and the same first term (minimal for E'). Note, in
considering the second of the conditions defining the order of couples,
that all the elements of E" are smaller than all the elements that
remain, which are in E' _E", which themselves are all smaller than
the elements of E-E'.
There is a sort of concentric embedding, where the couples of each
inner circle are smaller than those of the exterior boundary.
Consider, finally, in E", the property 'being an ordinal that figures
in second position in one of the couples in E"'. There is a minimal
ordinal for this property. But, this time, the set obtained consists of
one couple only. This is because, in E", the first term of the couples
is fixed (it is the minimal first term for the couples in E'). In fixing
the second term (as minimal for this place), one couple is entirely
determined. But the others are themselves smaller than all the couples
in E'-E", which are smaller than the couples in E-E'. So the minimal
couple obtained in E" is in fact minimal in E-QED.
This property of minimality for the order of couples of ordinals
grants us three essential freedoms:
1 Given a couple C, it is possible to designate the unique couple
that will come directly after it in the order we have defined. To
do so, it suffices to consider, in a suitable set that contains C, the
subset of those that are larger than it. This subset will have a
minimal element, which is the smallest one to be larger than C,
and is thus the 'successor' of C.
2 If a property of couples defines a set (the set of couples which
possess that property), then we can safely speak of the smallest
couple in that set, and therefore of the smallest couple to possess
that property.
3 Given a set of couples, we can speak of the upper bound of that
set, as we can for sets of ordinals (see 12.16): it suffices to consider
the smallest couple that is larger than all the couples in the set.
'Well-orderedness' allows thought to move between interior minimal-
ity and exterior maximality: the smallest of a given set, and the first
(outside) to be larger than all those in that set. The trap is to imagine
that one thereby gains access to internal maximality: this is not at all
the case because, for couples as for ordinals, that which goes to the
limit is not internally maximisable.
17.8. We are speaking of succession and limit. Here we return, let
us remark, to the disputations of chapter 9. Discovering the kinship
between ordinals and couples of ordinals was our latent motive.
17.9. Let's begin with an example. What can we say of a couple of
the form (W"S(W
), where WI is any ordinal whatsoever apart from
0, and where S(W
) is the successor of any ordinal W
? Everything
depends on the maximal ordinal in the couple. Suppose that WI is
maximal and thus that S(W
) E WI. If we compare the couple to all
others that also have WI as their maximal ordinal, we see that it is:
- larger than all those where W I only comes in second position in
the couple (primacy of first position, condition 2 of the ordering
of couples);
- larger than all those which, in second position, have an ordinal
smaller than S(W
) (third condition of order); in particular, it is
larger than the couple (W"W
- smaller than all those which have, in second position, an ordinal
larger than S(W
). In particular, it is smaller than the couple
))), supposing that S(S(W
)) remains smaller than WI.
thus leaving WI'S maximal status intact. But let's assume this
It seems clear from this that the couple (W
), given the
assumed hypothesis, intercalates itself exactly between the couple
) and the couple (W
))). More specifically, we can say
that it succeeds the first of the two couples.
If, on the other hand, we take the couple (W),L) where L is a limit,
and still suppose that WI is maximal in the couple, we cannot deter-
mine a couple that it succeeds. This couple is certainly larger that all
the couples of the form (W
) where W
is smaller than L (third
condition of order). But L, a limit ordinal, precisely does not succeed
any of the W
in question. There is therefore only one possibility: the
couple (WhL) is the upper bound (see N6) of the set of couples
), where W
E L, with WI, of course, being maximal - that
is to say, larger than L. We can also say that the couple (WhL) is the
limit of the couples (W
) for W
less than L.
Finally, take the couple (0,S(W
). The Max. of this couple is
evidently S(W 2). But it is certainly the smallest couple to have this
Max. In fact, its first term is minimal (it is 0), so every couple C
where Max(C) = S(W2) and where the first term is not ° -therefore
every couple of this sort other than our example - is greater
than it.
Being the smallest couple whose Max. is S(W
), our couple must
succeed the 'largest couple' - if it exists - whose Max. is immedi-
ately inferior. Note that these notions of 'larger' and 'immediately
inferior' can be disrupted by the intervention of limit ordinals. All
the same, this is not the case in our example: since the Max. of
our couple is S(W
), an immediately inferior Max. exists: it is W
What would be the largest couple whose Max. is W
? Obviously
that couple whose first term is maximal (condition 2 of order). But
the first term of a couple whose Max. is W
attains its maximum
when it is equal to W
• For, if it surpasses W
, the Max. changes.
So the couple that immediately precedes (0,S(W
) in the order of
couples is (W
). We can also say that (0,S(W
) is the successor
couple of (W
17.10. What we really want is to 'ontologise' couples of ordinals, as
we did for ordinals: find an intrinsic characterisation, not bound to
order alone, of successor couples and limit couples. The examples in
the previous paragraph will guide us.
17.11. Let's begin with couples containing 0.
We have remarked that couples of the form (O,W
), for all WI other
than 0, are the smallest ones whose Max. is WI' This allows us to
characterise them immanently:
1 A couple of the form (O,S(Wd) is always a successor (it succeeds
»). Thus the couple (0,1) is a successor (it succeeds the
minimal couple (0,0»).
2 A couple of the form (O,L) is always a limit: it is the upper bound
of the sequence of couples (WI,W
) where WI and Wz pass into
the limit ordinal L. So that the couple (0,00) is the limit of all the
couples (m,n) where m and n are finite ordinals (and therefore
natural whole numbers, see chapter 11).
Couples of the form (WhO) depend just as directly, as regards their
intrinsic characterisation, on the nature of the ordinal WI:
1 A couple of the form (S(W I ),0) is the smallest couple to have
S(Wd as Max. in first position. It is larger than all those which
have S(Wd as Max. in second position - that is, couples of the
form (W
,S(Wd) where Wz is smaller than S(Wd. In fact it
comes just after the largest of these couples, which latter will
evidently have the largest possible first term to conserve S(Wd's
status of maximum in second position. This largest first term is
WI. the immediate predecessor of S(Wd. The largest of the
couples which come before (S(W I ),0) is therefore the couple
(W"S(Wd). We can conclude: every couple of the form (S(WI),O)
is a successor. Thus the couple (1,0) is a successor (it succeeds
2 A couple of the form (L,O) is larger than every couple of the form
(W"L) where WI is less than L. But there is no such couple that
is larger than all the rest, because there is no W I that is 'closer'
to the limit ordinal L than all others (see 9.18). The couple (L,O)
is, moreover, smaller than all the couples of the type (L,W
) where
WI is not 0. In a sense, it makes a cut between the couples (W"L)
and the couples (L,O). All the same, amongst the latter there is a
minimal couple, which is the couple (L,l), and which therefore
succeeds (L,O). Here again we find the striking dissymmetry, char-
acteristic of the ordinals, between minimality (guaranteed) and
maximality (which presupposes succession). The couple (L,O) is
the limit, or upper bound, of the sequence (W"L) for WI E L,
and it immediately precedes the couple (L,l). It creates an infinite
adherence to its left, or 'on this side' of it, and the void of one
single additional step to its right, beyond it.
17.12. Let's now turn to 'homogenous' couples of the type (ShS2) or
). Everything will once more depend upon the Max. of these
• If 51 or LI are the Max., the problem is trivial: (s"sz) is a successor.
Just a moment of reflection will show that it comes just after the
couple constituted by 51 (the Max.) and the predecessor of 52. As
for (L
), it is surely the limit of the sequence of couples of the
type (L"W
), where WI traverses the elements of the ordinal L
• If Sz or L2 are the Max., things are not much more difficult. It is
certain that (51,52) is a successor: it comes just after the couple
constituted by the predecessor of 51 and by the Max. 52. As for
), it is assuredly the limit of the sequence of couples (W"L
where WI traverses the elements of L" from 0 'up to' L
17.13. We will finish with mixed couples. The method does not
change at all:
• If, in a couple of the type (s,L) or (L,s), it is L which is the Max.,
these couples are successors: they come just after the couples
obtained by replacing 5 with its predecessor.
• If 5 is the Max., the couples are limits of the sequences of couples
of the type (s,W
) or (W"s), where WI traverses the elements of
the limit ordinal.
17.14. Finally, we now have a table of immanent characterisations
of couples as follows:
Type Max. Example Character
Unique Special
(O,s) S (0,1) Successor
(O,L) L (0,00) Limit
(s,O) s (1,0) Successor
(L,O) L (00,0) Limit
(S"S2) Sl
(2,1) Successor
(1,2) Successor
(00,,00) Limit
(00,001) Limit
(s,L) s (S(oo),oo) Limit
(s,L) L (l,oo) Successor
(L,s) s (oo,S(oo) Limit
(L,s) L (00,1) Successor
This table has a perfect symmetry, broken only by the inaugural
couple of the void with itself, the ontological basis of the whole
17.15. It is entertaining to visualise the beginning ofthe sequence of
ordinal couples.
We have already seen that after the couple (0,0) comes the couple
(0,1), then the couple (1,0). One can quickly see that it is (1,1) that
succeeds (1,0), since it is the largest couple whose Max. is 1. Coming
next is (0,2), which is, as we have remarked, the smallest couple
whose Max. is 2. The readers can exercise themselves by calculating
the rest. If we draw the succession of couples onto a squared back-
ground, using the horizontal axis to represent the ordinal that occu-
pies the first place and the vertical to represent that which occupies
the second, we obtain the following:
<0,1> <1,0> <2,0> <3,0> <4,0>
What we see in this diagram is that the route through the couples
forms a kind of 'chain' which evidently could be projected onto an
ordinal axis. At any given moment we know how to 'produce' the
nth couple, as soon as its predecessor has been determined. It is
tempting to formalise this intuition by establishing a term-by-term
correspondence between ordinals and couples of ordinals, since we
have seen that the 'passage to the limit' represents no obstacle to our
doing so: there is a concept of the limit couple, structurally distinct
from the successor couple. This magnificent construction, which proj-
ects couples (representable on a plane or a surface) onto the linearity
of their constituents (single ordinals), is a triumph of ontology. It
shows that there is no more in the double than in the single. It lin-
earises the divergence of twoness.
17.16. Our underlying motive here is to show that couples of ordi-
nals behave 'like' ordinals themselves. The simplest way is to establish
between couples of ordinals and ordinals a biunivocal correspon-
dence (see 4.5). However, it is dubious, absurd even, to speak of a
correspondence or a function between two inconsistent multiplicities.
Neither the ordinals nor the ordinal couples are sets. How can we
justify comparing or linking these two untotalisable collections?
We have given the principle for the forcing of this impasse in
chapter 10: we must, if we can, define the correspondence between
the ordinals and the couples via transfinite induction, or recurrence.
The function will only be defined at successive levels, without us
having to consider the 'ails' between which it operates.
17.17. Let (((W"W
») be the function we wish to define and which,
to every couple of ordinals, will make correspond biunivocally an
ordinal: (((W"W2») = W
We are firstly going to root the function ( securely in its first value,
which will correspond to the smallest of the couples, the couple (0,0).
Refer back to chapter 10 for the whole of this procedure.
We posit explicitly:
We will then examine the case of successor couples (compare the
typology of couples in 17.14). Let C
be a couple which succeeds
couple C" which we will denote - in an extension of the notation
adopted for the ordinals - by C
= S(Cd. The simplest way is to make
correspond to the couple, via (, a Cl, which is the successor of couple
C" the successor ordinal of the ordinal which corresponds, via (, to
C,: we make the ordinals succeed 'in parallel' to the succession of
couples. We thereby respect the basic idea of induction, or recurrence:
supposing the function (to be defined for the couple C" we define it
by an explicit rule for the couple C
which succeeds C,. We therefore
) = ((S(Cd) = S(((Cd).
let's move on to limit couples. We suppose the function (to be
defined for all couples that precede a limit couple CL. To all these
couples, the function (makes correspond an ordinal W = ((C). The
idea is evidently to take, as value of ( for the limit couple Cl, the
ordinal that comes just 'after' all of the ordinals thus associated, via
(, with the couples that precede CL. We know of the existence of
this ordinal that comes just 'after' a set of ordinals (see N6): it is
the upper bound of that set, denoted by sup. We posit then that
((Cl) is the sup. of all the ordinals ((C) for the set of C smaller than
CL. So:
RULE 3 ((Cl) = sup.(((C)), for C < CL.
The inductive definition of ( is now complete, since we have
covered the three cases - the minimum (D,D)), successors and limits
- defining (via an explicit rule which makes its value depend upon
the values obtained 'below' the term in question.
17.18. A few examples.
What, for example, is the value of (((D,1))? We have seen that the
couple (D,l) is the successor, in the order of couples, of the couple
(D,D). We apply rule 2: (((D,l}) = S((((D,D})). But rule 1 indicates that
(((D,D}) = D. Then it must be: (((D,1}) = SID) = 1.
What is the value of (((D,ro})? We have seen that the couple (D,ro)
comes just after the set of all the couples (m,n), where m and n are
finite ordinals (the natural whole numbers). Now it is clear that (
associates a finite ordinal with each of these couples, since a successor
couple will be associated with the successor of the ordinal that cor-
responds to its predecessor, and since one begins from D. To couples
of the type (m,1t) will correspond the sequence D,1,2, etc. Conse-
quently, (((D,ro)) will have as its value the upper bound of all the finite
ordinals, that is, the first infinite (or limit) ordinal, which is to say roo
Thus (((D,ro}) = roo
These elementary examples demonstrate that we are indeed in a
position to calculate ( for any couple whatsoever: it is enough to
'progress' the length of the well-ordercdness of couples. The value
for the first couple being fixed, rules 2 and 3 allow us to know the
value of ( for a couple C on the basis of the values which ( assigns
to the couples which precede it.
17.19. That our function (, defined inductively with these three rules,
is definitely biunivocal merits verification, whatever evidence we may
already have on this point.
It must first of all be confirmed that (is injective, or, in Dedekind's
terms, distinct (see 4.5). In other words that, if couple C
is different
from couple C
, then ordinal ((Cd is different from ordinal ((C
). We
can assure ourselves of this by casting our eye over the rules of induc-
tion. If two couples are different, they are ordered; say, C
• The
value of ((C
) depends on the value of (for the couples which precede
it, and it is different from all of these values. Specifically, it is different
from the value of (for C
which comes before C
• We can therefore
be sure that C
((Cd...;:. ((C
). Function ( is injective.
In fact, we have a stronger property here: the function 'projects'
the order of couples into the order of ordinals (technically, it is a
homomorphism from the order of couples into the order of ordinals),
such that, if C
, then ((Cd E ((C
). For, if C
comes after C
value for ((which is either the successor of the value of the couple
which precedes it, or the upper bound of the values of (for all the
couples which precede it) in any case surpasses the value of (for C
Consequently: C
((Cd E ((C
It remains to be shown that function (is surjective, a modern word
meaning that every possible value of the function is effectively ful-
filled. In other words that, for every ordinal W, there exists an ordinal
couple C for which ((C) = W.
Suppose that an ordinal W exists whose value for function (is not
a couple C. Then there exists a smaller such ordinal (principle of
minimality), say w. Thus all ordinals smaller than w do correspond,
via, (, to a couple. We can see then that w must necessarily also,
contrary to the hypothesis, correspond, via (, to a couple. Because,
if w is a successor, which means that w = S(wd and ((C) = WI. it must
then be (rule 2 of the inductive definition of f) that ((S(C)) = S(wd =
w. And, if w is a limit, then, since all the ordinals which precede w
correspond, via (, to couples, w itself appears as the upper bound of
all those ordinals, and thus, from rule 3, its value for (will be the
couple that comes 'after' all the couples corresponding to ordinals
smaller than w.
So, finally, (is indeed a biunivocal correspondence between couples
and ordinals. This correspondence is, in addition, an isomorphism
between the structure of order of couples (via the Max., the first term,
and then the second term) and the structure of order of the ordinals
(belonging). Suffice to say at this point that the ordinal couples are
a sort of 'doubled' image of single ordinals. Taken 'two by two',
nature is still similar to itself. Nature is its own mirror.
17.20. These wanderings in nature through the looking-glass of the
double teach us:
- that there exists a well-orderedness over couples of ordinals, such
that these couples obey, as do ordinals, the principle of
- that we can speak, as for the ordinals, of successor couples and
limit couples, and that these attributes can be uncovered by imma-
nent examination alone of the structure of the couples which
possess them;
- that there exists between couples and ordinals a function ( which
has all the characteristics of a biunivocal correspondence, except
that the totalities between which this function operates are incon-
sistent, so that (must be defined by transfinite induction;
- that this function ( defines an isomorphism between the structure
of order of couples of ordinals and the structure of order of the
ordinals, so that C] < C
implies that ((Cd E ((C
In the mirror of the double, nature perseveres in all of its formal
Identical means would allow us to establish that triplets of ordi-
nals, of the form (WJ,W
), have the same properties as couples
do, and in particular that they are in biunivocal correspondence with
the single ordinals. The same goes for n-tuplets of ordinals of the
form (WbW
). In matter, it is only the first step that costs.
Doubled, nature maintains its order. Reduplicated in finite series as
long as you like, nature persists in maintaining its first identity. Stabil-
ity, homogeneity, order, minimality, the ontological hiatus between
successors and limits: all of this remains when the simplicity of the
ordinal is multiplied within the limits of the finite. Nature is its own
hall of mirrors.
17.21. Mallarme wrote: 'Nature is there, it will not be added to,.3
And it is a fact that, if one adds to nature, and even if one adds and
adds, and so on repeatedly, the domain of natural multiples attests
unabated to the pregnancy of the Same. This is what we grasp in
every experience of the natural: that ramified growth, reproductive
division, far from suggesting to us the Other, reposes in itself, in the
eternal seat of its order.
Now, we know that every operation, every algebra, is concerned
with a doubling or tripling of the terms upon which one operates.
We add two numbers to obtain a third, calculate the smallest
common divisor of two numbers, arrange in a finite sequence the
components of a polynomial ... All these disciplines of reckoning
and algebra have as their substructure a finite listing of numerical
If it is true that natural multiples, ordinals, furnish the matter of
Number, we can understand why the possibility of operations, of
algebra, of reckoning, finds its ontological guarantee in nature's
capacity to maintain the identical within division. Beneath the appar-
ent variegation of schemes of reckoning, the variety of operations
and of algebraic structures, lies this perseverance of natural being,
this immanent stability in finite seriality. An operation is never any-
thing more than the mode in which our thinking accords with
Mallarme's maxim: if, without exposing ourselves to the disintegra-
tion of the Other, we can combine two Numbers - 'add' one to the
other - it is because nature, taken as double, added to itself, re-
attached to itself, maintains the immanent form of the multiple-beings
through which it inconsists.
An operation, a counting, an algebra, are only marks of our
thought's being caught in the mirror-games which it pleases being
to proffer, under the law of the Same to which natural multiples
dispose it.
Algebra of Numbers
18.1. We must finally come to counting.
Once its being has been fixed, the combinatory capacity of Number
is a mere consequence. It arises from an investigative ingenuity as to
the ways in which couples or triplets of Numbers can be linked. But
the source of these links is held completely within the concept by
means of which Number is anchored in being. All that operations
can do is to deploy - in the numberless domain of Number - the
prodigality of being in its possible connections.
Concomitantly, the difficulty resides in the choice of 'good' defini-
tions of the links, so that they should conform to the facilities of
calculation: we wish the operations to be associative, for there to be
a neutral element, inverses, and it would help if they were also com-
mutative. We would be even happier if operations combined well-
behavedly amongst themselves, with a distributivity of one with
regard to the other. To arrive at these results, Number must be scru-
tinised and we must carefully authenticate the links we wish to
18.2. The substantial results to be obtained through the ingenuity of
operational definitions are as follows:
1 We can define a first operation on Numbers named addition and
denoted by +, which has the properties of a commutative group:
- associativity: Nt + (N
) = (Nt + N
) + N3 (one can count
'in any order', and achieve the same result);
- a neutral element (which is Zero): N, + 0 = N,;
- inversion (which is the symmetric counterpart): N, + (-N,) = 0;
- commutativity: N, + N2 = N2 + N,.
2 We can define on Numbers a second operation, named multiplica-
tion, and denoted by·, which has the following properties:
- associativity: N, . (N
- neutral element (which is the Number 1): N,·l = N,;
- existence of an inverse i(N) for every Number different from
Zero: N,·i(Nd = 1;
- commutativity: N,·N
3 Multiplication is distributive in relation to addition: N, . (N2 + N
These three operational considerations would lead us to say that
Numbers form a commutative {ield, if it were not for one problem:
Numbers do not even form a set, because they are inconsistent. How
can something be a field - which is supposed to be an algebraically
defined entity - if it cannot be counted as a multiple?
Therefore, prudently, we will say only this: that every set consti-
tuted of Numbers whose matter is less than a given cardinal infinity
(therefore every set constituted from Numbers whose matter is
bounded by a 'brute' fixed infinite quantity) can be given the structure
of a commutative field.
What is more, as can be proved for the
rationals or the reals, there are other sets of Numbers that are also
commutative fields. Numerical inconsistency can be 'sectioned' into
innumerable algebraic structures.
These logical caveats aside, the algebra of Numbers is the richest
conceivable: its calculative capacities equal- for example - those with
which the real Numbers furnish us (in particular, it can be proved
that every Number has a square root, which is not the case if one is
operating, for example, in the field of rational numbers).
18.3. A result at once laborious (in its procedures of verification) and
of key importance (for the validity of our concept of Number) is the
following: operations defined on Numbers coincide with operations
defined on 'our' numbers, if the latter are thought in their being as
Numbers. In other words: take two real numbers r, and r2, taken in
their usual algebraic sense. If the sum of rl and r2, such as we know
it, is the real number r3, then the sum of r, and r2 such as it is repre-
sented 'itself' in Numbers (with Numbers of matter finite or equal to
0), see 16.28) - 'sum' being taken in the sense of the addition defined
on Numbers - will be precisely the representative, within Number,
of the number r3. The same will go for multiplication, etc. More
technically, we can say that the field of the reals, as we know it in
classical analysis, is isomorphic to the reals thought as subset of
It is not, therefore, solely in their being that 'our' usual numbers
can be thought of as singular types of Numbers, but also in their
algebra. Our real Numbers are ultimately indistinguishable from real
numbers. In particular, real Numbers constitute a complete ordered
Archimedean field, which is the univocal customary definition of real
It can be said, ultimately, that all the dimensions and capacities of
'historical' numbers are retained by their presentative instance in the
innumerable swarm of Numbers. Which confirms:
- that the ontological essence of a number is nothing more than
that which our thought apprehends it to be when it is determined
as a type of Number;
- that the operational or algebraic properties are only the effect of
a correct determination, on the basis of natural multiplicities, of
the being of Number.
We therefore find the programme of unification of the concept of
Number (one sole concept which subsumes the natural whole
numbers, the negative whole numbers, the rationals, the reals and the
ordinals) to be wholly realised, firstly in multiple-being, and then in
the operational dimensions.
It is now possible for us to speak freely of, and to submit to cal-
culation, entities previously devoid of any sense, like the sum of an
ordinal and a real number, or the division of a transfinite ordinal by
a rational number, or the square-root of the division by three of an
ordinal, etc. Incredible equations like:
- which, in the dispersed and lacunary historical theory of numbers,
would have made absolutely no sense - in the unified framework of
the concept of Number become perfectly meaningful algebraic formu-
lae, indicating certain procedures of calculation and definite results.
Number thus founds in being the literal connection of what,
under the disparate name of 'numbers', had defined heterogeneous
18.4. The definition of operations on Numbers is essentially a techni-
cal affair. Whoever wishes to follow it in all its detail is referred to
the Iiterature.
Nevertheless, its animating spirit allows a revision of
concepts, a final passage through the Idea of Number. In particular,
the systematic use of transfinite induction highlights the fact that
Number, thought of in its being, is essentially an infinite multiple (the
section of a form from an infinite ordinal-matter). In the same way,
the recourse to sub-Numbers of a Number in order to construct
operations 'from below' attests to the importance of the fact that
every Number can be presented as a cut of its low set and its high
set (see chapter 14). And again, it is by presenting the result of an
operation as a cut (see chapter 15) - that is, by utilising the funda-
mental theorem - that we can handle induction. Lastly, the correla-
tion explored in chapter 17 between couples of ordinals and ordinals
plays a major role in this whole process - as one might expect (since
an operation connects two Numbers). So as not to forego these reca-
pitulations in thought, we will cover the essentials of the definition
of addition.
18.5. The general idea is as follows: given two Numbers NI and N
we can make them correspond to two ordinals W I and W 2 simply
by taking their respective matters, WI = M(Nd and W
). We
know that a certain ordinal corresponds to these two ordinals via
the biunivocal function (, which associates an ordinal with every
couple (Wl,W
) of ordinals (see 17.17). This ordinal will fix the
'level' of definition of the additive operation: we will suppose that
addition is defined for all couples of Numbers N3 and N4 of matter
W3 and W
, such that the ordinal that corresponds via (to the couple
) is smaller than the ordinal associated with the couple
). We then propose an explicit rule, which will define the sum
N I + N2 on the basis of sums of the type N3 + N
, defined at a lower
ordinal level.
Now, such sums are given by the sub-Numbers of NI and N
sub-Number, being a 'partition' of a Number for an ordinal smaller
than its matter, has an inferior matter.
We can then pass on to the next stage, which is the core of the
construction. Take the Numbers NI and N
, of matter WI and W
Consider a sub-Number N I/W3 of N I, and a sub-Number Nz/w4 of
N2. Now take the couples (W3,W
), or (WhW4)' I say that they are
lower than the couple (WI,W
), by the rules of order of couples, and
remembering that W3 and W4 are respectively smaller than WI and
(see 17.6). This is an excellent exercise for the reader, but see
the note.
As the function f is an isomorphism of the order of couples with
the order of the ordinals, we will also have:
f(W3,WZ») E f(Wt.Wz») ,
f(Wt.W4») E f(W"Wz).
Which is to say that the ordinal level associated with couples
of Numbers of the type (Ndw3,Nz), or (NhNz/W4) will always be
lower than the ordinal level associated with couples of Numbers
Given this fact, in order inductively to define the sum of N, and
, we can suppose defined sums of the type Ndw3 + N
, or N, +
Nz/w4, which pertain to a lower ordinal level. We will thus pass on
to the definition of N, + N2 by formulating a rule which assigns the
value of this sum on the basis of the various values between N, and
on the one hand, the sub-Numbers of N, and N
on the other.
The immanent concept of sub-Number will serve to underwrite the
induction, which fixes their ordinal level on the basis of a couple
formed of the matters of the two Numbers under consideration.
Finally, the strategy will mobilise the fundamental theorem of the
cut. We will begin with the low set and the high set of the two
Numbers N, and N
• We suppose defined the sums of each of the
two Numbers with the sub-Numbers of the low set and of the high
set of the other Number, according to a fixed combination. These
sums can be assumed, because their ordinal level is lower. We can
thus obtain two sets of Numbers, and the sum of N I and N
will be
the unique Number defined as cut of these two sets.
18.6. Inductive definition of the addition of two Numbers
'Level zero' of the induction contains only the Number (0,0). It is the
only one to have ° as matter. We can thus posit:
We will now suppose that addition is defined for all levels lower
than an ordinal W, that is, all levels corresponding to Numbers N3
and N4 (taken in that order) such that, their respective matters being
W3 and W
, it is the case that f(WJ,W
Now take a couple of Numbers NI and N
such that, their respec-
tive matters being WI and Wz, it is the case that f(W"W
) = W. In
other words a couple of Numbers belonging to ordinal level W.
We have remarked that all the couples of type N, and Nz/w, or
N /w and N
, where N /w and Nz/w are sub-Numbers of N, and N
belong to ordinal levels inferior to those of the couple N, and N
and therefore inferior to W.
It follows that we can suppose defined all the additions of the type
N, + Nz/w, or Ndw + N2•
We must agree on an important written convention here. We will
write N, + Lo(N
) for the set of Numbers constituted by all the results
of the addition of N, with each of the Numbers of the low set of N
(the low set of N2 is constituted, remember, of all sub-Numbers of
smaller than N
). If Lo(N
) is empty, the Number denoted by
N, + Lo(N
) would be undefined (we will not consider this in the
calculations) .
In the same way, we write NJ + Hi(N
) for the set of Numbers
constituted by all the results of the addition of N, with each of the
Numbers of the high set of N2 (the high set of N2 being constituted
by all sub-Numbers of N
larger than N
). The convention will always
be not to bother writing this if Hi(N
) is empty.
We will adopt the same notation to designate sets of Numbers which
result from additions implicated in Lo(N,) + N
, or Hi(N,) + N
Addition will then be defined as follows: on the one hand we take
the set of Numbers constituted by all the Numbers of Lo(Nd + N2,
together with all the Numbers of N, + Lo(N
); on the other hand,
the set constituted by all the Numbers of Hi(Nd + N
, together with
all the Numbers of N
+ Hi(N
). In other words, we 'collect' on one
side the Numbers which are the sum of N, and N2 and the low sub-
Numbers of the other Number, and on the other side the same sums,
but with the high sub-Numbers.
We thus obtain two sets of Numbers, which we can call Land H.
It is not hard to prove, by way of a 'incremental' induction which
I leave to one side,4 that Land H are in a situation of a cut: every
Number of L is smaller than every Number of H.
We then utilise the fundamental theorem (chapter 15). The result
of the addition of N, and N2 will be precisely the Number which
makes a cut between these sets, that is, the unique Number of minimal
matter situated between the sets:
We posit:
L = (Lo(Nd + N
,N, + Lo(N
H = (Hi(Nd + N
,N, + Hi(N2))
RULE 2 N, + N2 = UH, cut of the two sets defined above.
18.7. Addition is commutative
In fact, this cut, which supposedly defines the sum N2 + Nh operates
on the same sets as the cut which defines N I + N
, as one can show
inductively with no difficulty.
It is true at level 0, where there is only the sum, certainly commu-
tative, 0 + O.
Suppose that the sums of ordinal levels inferior to ((W
are commutative. Then, in particular, the sums Lo(Nd + N
or N, +
) are commutative. So the set L = (Lo(Nd + N
+ Lo(N
which serves to define N, + N
, is composed of the same Numbers
as the set L' = (Lo(N
) + (Nd,N
+ Lo(Nd), which serve to define
. Evidently, the same goes for the set H. And consequently,
, being defined by the same cut as NI + N
, is equal to it:
addition is commutative.
18.8. The Number 0, which is more precisely the Number (0,0), is
the neutral element for addition
It is a question of proving that, for every Number N, N + 0 = N.
Induction can this time be applied directly to the ordinal-matter of
the Numbers.
It is true at level 0, since rule 1 prescribes that 0 + 0 = O.
Suppose that this is true for all the Numbers of lower matter than
WI' In other words, for every Number N of matter w such that
Now take a Number NJ of matter WI' Let's examine the sum
N I + O. The sets Land H of the cut which define the addition
L = (Lo(Nd + 0, Lo(O) + Nd
H = (Hi(Nd + 0, Hi(O) + Nd
But the low set and the high set of the Number 0 - that is, (0,0) -
are empty (0 has no sub-Numbers). The conventions adopted in 18.6
prohibit us from taking into account the terms Lo(O) + NI and
Hi(O) + N,. So we actually have:
L = (Lo(Nd + 0)
H = (Hi(Nd + 0)
But Lo(Nd and Hi(Nd are composed of sub-Numbers of Nh
and therefore of Numbers of lower matter than WI' Consequently,
the hypothesis of induction applies to all the Numbers of Lo(N d
or of Hi(Nd: for any such Number, say Ndw2' it is the case that
Ndw2 + 0 = N11w2'
We can, with a slight abuse of notation, write this result in the
form Lo(Nd + 0 = Lo(Nd, Hi(Nd + 0 = Hi(Nd. So that, ultimately,
Land H, which define by a cut the sum N 1 + 0, are no other than
Lo(Nd and Hi(Nd. But the Number defined by a cut between its low
set and its high set is precisely the Number N 1 itself, and so it is
indeed the case that N I + 0 = N I'
The induction is complete: for a Number N, whatever its matter,
o is a neutral element for addition.
18.9. Every Number N apart from 0 allows the Number -N as its
inverse for addition: N + (-N) = 0
An important point: since -N inverts the form and the residue of N,
the low set of -N is composed of the Numbers -Nlw, where Nlw is a
Number from the high set of N; and the high set of -N is composed
of the Numbers -Nlw, where Nlw is a Number from the low set of N.
A sub-Number Nlw is in the low set if w is in the form, and it is in the
high set if w is in the residue. These determinations will be inverted in
-N. And, since everything that precedes w in N is also inverted (what
was in the form is in the residue, and what was in the residue is in the
form), in addition to the exchange of the low set and the high set, we
will also have an exchange of the signs of positive and negative.
In an abuse of notation, we could therefore write the high set of
-N as -(Lo(N)), and the low set of -N as -(Hi(N)).
The result (see the inductive definition of addition) is that the two
sets Land H which define by a cut the sum N + (-N) are the
L = (Lo(N) + (-N), N + (-(Hi(N))))
H = (Hi(N) + (-N), N + (-(Lo(N))))
So the strategy of the proof consists in proving that all the N
umbers of L are negative and all the Numbers of H positive. The
result is that 0 is situated between Land H and that, being necessarily
of minimal matter in that position, it is 0 that occupies the position
of the cut between Land H. Consequently, N + (-N) = O.
LEMMA If the sum NI + N2 is positive, if NI + N2 > 0, then
-(Nd < N
, and -(N2) < N
The lemma is true at ordinal level 0, because at that level it cannot
possibly be the case that N 1 + N2 > O.
Suppose that it is true up to ordinal level W: for every pair of
Numbers N3 and N4 such that {(W
) E W, the property in ques-
tion holds. I say that it also must hold for every pair of Numbers N,
and N2 such that {(W"W2) = W.
The sum N, + N2 is defined by the cut UH. If this cut is
positive, it is because set L contains positive Numbers,s or else the
cut would be negative or null (see the argument on cuts in 15.11).
As for set H, it only contains positive Numbers. Consequently,
there are Numbers in Lo(Nd + N2 or in N, + Lo(N
) that are
positive, and all the Numbers of Hi(Nil + N2 or of N, + Hi(N2)
are so.
Take for example N,lw + N2 as a positive Number of Lo(Nd +
. The pair of N,lw and N2 is of lower ordinal level than W, and
the lemma is therefore supposed to be true of it: since the sum N dw
+ N2 is positive, it is the case that -(N2) < N
1w, and, since N dw is
in the low set of N" it is a fortiori the case that -(N2) < N,. In exam-
ining the other components of sets Land H, the lemma can be
established in all generality.
Now let's come back to the sum N + (-N). Consider the set L
which defines it by a cut, so:
L = (Lo(N) + (-N), N + (-Hi(N))).
Suppose that there are positive Numbers in L. Take for example
one such Number Nlw + (-N), where Nlw is from the low set of N.
In virtue of the lemma, it is the case that -(-N) < Nlw, so N < Nlw,
which is impossible since Nlw, being from the low set of N, must be
smaller than N. If N + (-N/w) is positive, N/w being in the high set
of N, it must be the case that N/w < N, which is prohibited, since
Ndw belongs to the high set. We meet with an impasse, and so there
are no positive Numbers in set L.
Symmetrical deductions would demonstrate that there are no nega-
tive or null Numbers in set H.
Finally, the cut LlH which defines the result of the addition
N + (-N) operates between a set L of negative Numbers and a set H
of positive Numbers. The N umber of minimal matter situated between
these two sets is necessarily 0, and so N + (-N) = O.
So we can say that -(N) is the inverse of N for addition.
18.10. Confirming that the addition of Numbers is associative is,
as always, a tiresome calculation. It is, it is ... To the extent that
we have established that Numbers, endowed - so to speak - with
addition defined inductively by the cut:
would form (were it not for the inconsistency of their 'All' ... ) an
ordered commutative group, of which the Number Zero (either (0,0)
or 0, it's all the same) is the neutral element.
To confirm that the 'representatives' in our Numhers of whole
positives and negatives, rationals, rea Is, ordinals, are in fact these
numbers themselves, but thought in their being, we must prove that
addition (in the normal sense) of these numbers coincides with addi-
tion of their being as Numbers. For example, if rl and r2 are numbers
from the real field, and if rl + r2 = r3 with 'classic' addition, then the
Numbers rJ, r2 and r3 defined as Numbers of finite matter or of matter
(0, as we presented them in 16.27, are such that, addition being
defined inductively as above, it is always the case that rl + r2 = r3.
These confirmations of algebraic isomorphy demand no little inge-
nuity, above all when it comes to multiplication (which labyrinth we
will avoid entering into).
18.11. I will content myself with carrying out the verification for
natural whole numbers.
Remember (from 16.13) that a natural whole number n presented
as Number is of the form (n,n), where n is a finite ordinal. Recall
also (ibid.) that the low set of n is constituted by all the whole
Numbers lower than n, and that its high set is empty.
Take two natural whole Numbers (nJ,nd and (n2,n2)' Their sum is
formally defined by the cut:
But, as Hi(nd and Hi(n2) are empty, the sums of set H of the cut
are not defined (convention on the definition of addition, see 18.6).
Set H is therefore empty, which amounts to saying that the sum is
simply the upper bound of set L.
Since Lo(nd is the set of Numbers lower than nJ, the sum Lo(nd
+ n2 is constituted by all the sums ° + n2, 1 + n2, ... (nl - 1) + n2'
And, in just the same way, nl + Lo(n2) is constituted by all the
sums nl + 0, nl + 1, ... , nl + (n2 - 1).
The largest Number of these sums is in all evidence the Number
nl + n2 - 1.
Reasoning by induction: suppose that, up to the ordinal rank
which corresponds to the pair of Numbers nJ,n2 (so, in reality, the
Numbers (nJ,nd and (n2,n2)), therefore up to the ordinal w = f(nJ,n2),
it is true that the sum of wholes as Numbers will be the Number
which corresponds to the (normal) sum of the wholes. In particular,
that it is true for the pair nl>(112 - 1), which is evidently of a lower
ordinal rank than the pair nl>n2. It is therefore to be supposed that
(nl>nd + ((n2 - 1),(n2 - 1)) = the Number which corresponds to the
ordinary addition of the numbers n, and (n2 - 1), that is, the Number
(n, + (n2 - 1),n, + (112 - 1)), where the sign + denotes the ordinary
addition of whole numbers.6
Now we come to see that the largest Number in the set L which
defines n, + n2 is precisely n, + (112 - 1). In virtue of the hypothesis
of induction, this Number is the Number which corresponds to its
being written as an ordinary addition - the Number which inscribes
the whole number n, + n2 - 1.
Now, n, + n2 (in the sense of the addition of Numbers) is the upper
bound of L. And every upper bound is a Number of the type (W,W),
as we have demonstrated in 15.9. The upper bound of L will therefore
be the smallest Number of the type (W,W) to be superior to the largest
Number in L, which is the Number (n, + n2 - 1,11, + n2 - 1) (where
the signs + and - have their traditional meaning, as when dealing
with numbers). This Number is evidently (n, + n2,n, + n2), because
n, + n2 is the finite ordinal which comes immediately after the finite
ordinal n, + n2 - 1.
Consequently, the sum (in the Number sense) of the two whole
Numbers n, and n2 is the Number that represents the number sum
n, + n2 (in the number sense). The addition of whole Numbers is
isomorphic to the traditional addition of whole numbers.
The treatment of whole negative numbers poses no great problem
(an interesting exercise). Thus it is confirmed that the whole positive
and negative Numbers form a commutative group isomorphic to the
additive group of the ring 7L. of the algebraic whole numbers.
The reader will have grasped the essence of operational proce-
dures: find a 'good' inductive definition of the links, prove the classic
algebraic properties (associativity, commutativity, neutral element,
inverse, distributivity ... ), confirm that what one obtains is isomor-
phic, for the classical numbers represented in Numbers, to the
structures which these numbers are endowed with.
However laborious these efforts might be, they lead to the desired
conclusion: all the classic algebraic structures (the ring 7L. of algebraic
whole numbers, the field a of rationals, the field IR of reals), and all
the 'inconsistent' algebras (addition and multiplication of ordinals)
are isomorphic to the substructures discernible within Numbers.
And so it is that all types of numbers, without exception and
in their every dimension, are subsumed by the unique concept of
In Conclusion:
From Number to Trans-Being
19.1. Number is neither a trait of the concept, nor an operational
fiction; neither an empirical given, nor a constitutive or transcenden-
tal category; neither a syntax, nor a language game, not even an
abstraction from our idea of order. Number is a form of Being. More
precisely, the numbers that we manipulate are only a tiny deduction
from the infinite profusion of Being in Numbers.
Essentially, a Number is a fragment sectioned from a natural mul-
tiplicity; a multiplicity thought, as ordinal, in its being qua being.
The linear order of Numbers, like their algebra, is our way of tra-
versing or investigating their being. This way is laborious and limited.
It exhibits Number in a tight network of links, whose three principal
categories are succession, limit and operations. This is where the illu-
sion arises of a structural or combinatory being of Number. But, in
reality, the structures are consequences, for our finite thought, of that
which is legible in Number as pure multiplicity. They depose Number
in a bound presentation which makes us believe that we manipulate
it like an object. But Number is not an object. Before every bound
presentation, and in the un-bound eternity of its being, Number is
available to thought as a formal section of the multiple.
We might also say that between Number, which inscribes its
section in the unrepresentable inconsistency of natural multiples, and
number, which we manipulate according to structural links, passes
the difference between Being and beings. Number is the place of the
being qua being, for the manipulable numericality of numbers.
Number ek-sists in number as the latency of its being.
19.2. This only makes it more remarkable, then, that we can have
some access to Number as such, even if this access still indicates an
excess: that of being over know ledges, an excess manifest in the
numberless extent of Numbers, compared to what we can know by
structuring the presentation of types of numbers. That mathematics
allows us at least to designate this excess, to accede to it, confirms
the ontological vocation of that discipline. The history of mathemat-
ics, for the concept of Number as for every other concept, is precisely
the history, interminable in principle, of the relation between the
inconsistency of being as such, and what our thought can make con-
sistent of this inconsistency. Mathematics establishes ontology as the
historical situation of being. It progresses constantly within ontico-
ontological difference, bringing to light, as the latency of the struc-
tures presented in the ontological situation, an excessive horizon of
inconsistency, of which structures are only effects for a finite thought.
It is this trajectory which we have reconstructed at one of its points:
that which designates, beyond numbers, the inconsistent multiple-
eternity of Numbers.
19.3. Number is thus rendered over to being, and subtracted from
the humanity of operations or figures of order, which nevertheless it
continues to subtend in thought. The task concerning Number, and
numbers, can only be to pursue the deployment of their concept
within ontico-ontological difference. Number falls within the exclu-
sive purview of mathematics, at least so far as the thinking of number
is concerned. Our philosophical project prescribes this exclusivity,
and designates where Number is given as the resource of being within
the limits of a situation, the ontological or mathematical situation.
We must abandon the path of the thinking of Number followed
by Frege or Peano, to say nothing of Russell or Wittgenstein. We
must even radicalise, overflow, think up to the point of dissolution,
Dedekind's or Cantor's enterprise. There exists no deduction of
Number, it is solely a question of a fidelity to that which, in its
inconsistent excess, is traced as historical consistency in the intermi-
nable movement of mathematical refoundations.
The modern instance of this movement attests to the void and the
infinite as materials for the thinking of Number. Nevertheless, none
of these concepts can be inferred from experience, nor do they propose
themselves to any intuition, or submit to any deduction, even a tran-
scendental one. None of them amounts to the form of an object, or
of objectivity. These concepts arise from a decision, whose written
form is the axiom; a decision that reveals the opening of a new epoch
for the thought of being qua being. Being asks nothing more of us,
at this point, than that we doggedly pursue the inscription - within
a revised ontological situation - of that which, in tracing the incon-
sistent latency of being, faithfully prepares the rupture at a point of
that place where it consists for us.
19.4. It is then possible to maintain that the contemporary 'banalisa-
tion' of number is outside all thought. The reign of number, the
portents of which I discussed at the beginning of this book, is intran-
sitive to the mathematical thought of Number. It imposes the falla-
cious idea of a bond between numericality and value, or truth. But
Number, which is an instance of being as such, can support no value,
and has no truth other than that which is given to it in mathematical
thought, effectuating its historical presentation for us.
If the reign of number - in opinion polls or votes, in national
accounts or in private enterprise, in the monetary economy, in the
asubjectivising evaluation of subjects - cannot be authorised by
Number or by the thinking of Number, it is because it follows from
the simple law of the situation, which is the law of Capital. This law
assures, as does every law, the count-for-one of that which is pre-
sented in the situation, it makes our historical situation consist, but
it cannot make any claim to truth: neither to a truth of Number, nor
to a truth which would underlie that which Number designates as
form of being.
In our situation, that of Capital, the reign of number is thus the
reign of the unthought slavery of numericality itself. Number, which,
so it is claimed, underlies everything of value, is in actual fact a pro-
scription against any thinking of number itself. Number operates
as that obscure point where the situation concentrates its law;
obscure through its being at once sovereign and subtracted from all
thought, and even from every investigation that orients itself towards
some truth.
The result is that all thought necessarily deploys itself today in a
retreat with regard to the reign of number, including every thought
that tries to make a truth of Number. It is in this sense that we must
hearken to Mallarme's slogan, more pertinent than ever: that of
restrained action.
This whole meditation on the concept of Number, because it
restores it to being, necessitates the inversion of the contemporary
judgement such as it is presented under the banner of number. We
must say, against this judgement, that nothing made into number is
of value. Or that everything that traces, in a situation, the passage of
a truth shall be signalled by its indifference to numericality. Not so
that this indifference can in its turn be made into a criteria, because
many projects, having no number, have no truth either. But this indif-
ference is a necessary subjectivity.
The reverse side of the abundance of capital is the rarity of truth,
in every order where truth can be attested to: science, art, politics
and love.
19.5. But if the true is, on principle, in our situation, subtracted from
the reign of number, which is only a law of this situation, what is
the origin of this process?
A truth can depend neither upon being as such (this is why it does
not signal itself through a Number) nor upon the contemporary situ-
ation, which is that of Capital (this is why it does not signal itself
through numbers). Its origin is evental. But the event is not non-being,
however much it exceeds the resources of situation-being. The best
way to say it would be that the event is of the order of trans-being:
at once 'held' within the principle of being (an event, like everything
that is, is a multiple) and in rupture with this principle (the event
does not fall under the law of the count of the situation, so that, not
being counted, it does not consist). Evental trans-being is at once
multiple and 'beyond' the One - or, as I have chosen to call it, ultra-
One. The possibility that there can be a truth, in a situation whose
state has wholly succumbed to numbers, depends upon a fidelity,
subtracted from numbers, to this ultra-One.
To think Number, as we have tried to do, restores us, either
through mathematics, which is the history of eternity, or through
some faithful and restrained scrutiny of what is happening, to a
supernumerary hazard from which a truth originates, always heter-
ogenous to Capital and therefore to the slavery of the numerical. It
is a question, at once, of delivering Number from the tyranny of
numbers, and of releasing some truths from it. In any case, restrained
action is the principle of a remote disorder: it establishes mathemati-
cally that order is but the all-too-human precarity of a thinking of
the being of Number; it proceeds, effectively and theoretically, to
the downfall of numbers, which are the law of the order of our
'Like a god, I put in order neither one nor the other .. .'2
Translator's Preface
1 London: Continuum, 2005.
2 Minneapolis: University of Minnesota Press, 2003.
3 London: Continuum, 2004; in particular, ch. 5, 'The Being of
Number', represents an extremely condensed gloss of the present
4 See for example ch. 2n 4.
Chapter 0 Number Must Be Thought
1 [Dedekind, R., Was sind und was sol/en die Zahlen (Braunschweig:
F. Vieveg, 1888); English translation The Nature and Meaning of
Numbers, in Beman, W. W. (ed., trans.), Essays on the Theory of
Numbers (La Salle, IL: Open Court, 1901; reprinted NY: Dover
1963). Badiou's reference is to the translation by J. Milner, with H.
Sinaceur's introduction, Les Nombres, que sont-i/s et a quoi servent-
ils? (Paris: Navarin, 1979). All references given below are to the
numbered sections of Dedekind's treatise. - trans.)
2 [aei 0 anthropos arithmetizei - 'man is always counting' - in
Dedekind, 'Numbers', Preface to the first edn. - trans.)
3 [Marx and Engels, Communist Manifesto, translated by S. Moore,
with introduction and notes by G. S. Jones (London: Penguin,
2002), p. 222. - trans.)
Chapter I Greek Number and Modern Number
1 Consider, for example, the definition of number in Euclid's Elements
(Book VII, definition 2): EO''tlV 'to h: 11OvuBcov O'UYKEtIlEVOV
We might translate it as follows: 'A number is a multitude
composed of unities.' The definition of number is secondary, being
dependent upon that of unity. But what does definition 1, that of
unity, say? Movu£ EOTLv, EKaOTOV TUN OVTCOV EV AEYETOl:
'Unity is that by virtue of which each being is said to be one.' We
can see immediately what ontological substructure is presupposed
by the mathematical definition of number: that the One can be said
of a being in so far as it is.
2 [trait: see ch. 2n 3. - trans.]
3 [Conway, J. H., On Numbers and Games (London Mathe-
matical Society Monographs no. 6, London: Academic Press, 1976).
- trans.]
4 [Knuth, D. E., Surreal Numbers (Reading, MA: Addison-Wesley,
1974). - trans.]
5 [Gonshor, H., An Introduction to the Theory of Surreal Numbers
(London Mathematical Society Lecture Note Series, 110, Cam-
bridge: Cambridge University Press, 1986). - trans.]
6 [Since the language of 'whole' and 'natural' numbers is informal
and not always applied consistently, it is worthwhile to set out the
usage of the present work, along with the formal mathematical
• whole numbers: 0, 1, 2, 3 ... (the non-negative integers, Z*).
• natural whole numbers: 1, 2, 3 ... (the positive integers, Z+).
• 'relative' whole numbers: .. . -3, -2, -1, 0, 1,2,3 ... (the inte-
gers, Z). - trans.]
7 On the dialectic - constitutive of materialist thought - between
algebraic and topological orientations, the reader is referred to my
Theorie du Sujet (Paris: Seuil, 1982), pp. 231-49.
8 [See Bourbaki, N., Elements de mathematique, Livre 1: Theorie des
ensembles (Paris: Hermann, 1954); English edn Elements of Math-
ematics, Vol I: Theory of Sets (Reading, MA: Addison-Wesley,
1968). - trans.]
9 The theme of the cut, in its concept and its technique, is treated in
chapter 15 of this book.
10 [See Dedekind, Numbers, §73. - trans.]
11 [Frege, Gottlob, Die Grundlagen der Arithmetik: Eine logisch-
matematische Untersuchung uber den Begriff der Eahl (Breslan,
1884); The Foundations of Arithmetic, English translation by J. L.
Austin (2nd revised edn, Oxford: Blackwell, 1974). References given
below are to the numbered sections of Frege's text. The first German
edn appeared in 1884. - trans.]
12 [See Frege, Foundations, §28-§29. - trans.]
13 For a particularly brief introduction to the different types of numbers
which are used in modern analysis, refer for example to J. Dieudon-
ne's Elements d'analyse, I: Fondements de ['analyse moderne (Paris:
Gauthier-Villars, 3rd edn, 1981), chs 1-4.
14 [unique nombre qui ne peut pas hre un autre: From Mallarme's
'Un Coup de des jamais n'abolira Ie hasard', translated in E. H.
Blackmore and A. M. Blackmore, Collected Poems and Other Verse
(Oxford: Oxford University Press, 2006), pp. 161-81 as 'A dice
throw at any time never will abolish chance' (translation modified).
15 Natacha Michel proposes the distinction between 'first modernity'
and 'second modernity' in L'Instant persuasif du roman (Paris: Les
Conferences du Perroquet, 1987).
16 Dedekind, Numbers, §64.
17 I give a detailed commentary on the Hegelian concept of number - a
positive virtue of which is that, according to it, the infinite is the
truth of the pure presence of the finite - in meditation 15 of L'Etre
et l'evenement (Paris: Seuil, 1988), pp. 181-90 [pp. 161-70 in
Oliver Feltham's English translation Being and Event (London:
Continuum, 2005). - trans.]
18 [See Frege, Foundations, §84-§86. - trans.]
19 [Dedekind, Numbers, §2. - trans.]
20 [Frege, Foundations, §74. - trans.]
21 [Dedekind, Numbers, §73. - trans.]
22 [Tout. - trans.]
23 [Dedekind, Numbers, §66. - trans.]
Chapter 2 Frege
1 The key text for Frege's conception of number is The Foundations
of Arithmetic [on which see above, ch. In 11 - trans.]. The funda-
mental argument, extremely dense, occupies paragraphs 55 to 86
(less than thirty pages in the cited edition). We must salute Claude
Imbert's excellent work, in particular her lengthy introduction.
[Badiou refers to Imbert's translation Les Fondements de
l'arithmhique (Paris: Seuil, 1969). - trans.]
2 [toute pensee emet un coup de des: Mallarme, 'Coup de des', p. 181.
- trans.]
3 [See Frege, Foundations, §§ 46-53. Badiou renders Frege's Eigen-
schaft as trait: Although Austin has 'property,' I have used 'trait'
here, so as not to lose the distinction present in Badiou's text
between trait and propriete. - trans.]
4 [In Austin's English translation, Frege's term Gleichzablig is ren-
dered as 'equal', but see Austin's note at §67 on possible alterna-
tives: I follow both Austin's advice and Badiou's use of equinumericite
in employing equinumerate as the term which avoids at once impre-
cision and ugly neologism. - trans.]
5 [Frege, Foundations, §68. - trans.]
6 [Ibid., §74. - trans.]
7 [Ibid., §77. - trans.]
8 [Ibid. - trans.]
9 [Ibid., §74. - trans.]
10 [Ibid. - trans.]
11 [Ibid. - trans.]
12 The letter (written in German) in which Russell makes known to
Frege the paradox that would take the name of its author is repro-
duced in English translation in From Frege to Godel, a collection
of texts edited by J. van Heijenoort (Cambridge, MA: Harvard
University Press, 4th edn, 1981) p. 124. Russell concludes with an
informal distinction between 'collection' [or 'set', German Menge-
trans.] and 'totality': 'From this [the paradox], I conclude that under
certain circumstances a definable collection [Mellge] does not form
a totality.'
13 Zermelo develops his set-theoretical axiomatic, including the
axiom of separation, which remedies Russell's paradox, in a 1908
text written in German. It can be found in English translation in
van Heijenoort's collection, cited in the preceding note. It comes
from Investigations in the Foundations of Set Theory, and espe-
cially its first part, 'Fundamental Definitions and Axioms', pp.
14 [Frege, Foundations, §58. - trans.]
15 The subordination of the existential quantifier to the universal quan-
tifier means that, given a property P, if every possible x possesses
this property then there exists an x which possesses it. In the predi-
cate calculus: 'v'x(P(x)) 3x(P(x)). The classical rules and axioms
of predicate calculus permit one to deduce this implication. Cf. for
example E. Mendelson's manual Introduction to Mathematical
Logic (NY: Van Nostrand, 1964), pp. 70-1.
16 [TO yag alho voEi:v taTLv TE Kat ElvUL, from Parmenides' poem.
- trans.]
Chapter 3 Additional Note on
a Contemporary Usage of Frege
1 Miller's text appears in Cahiers pour l'analyse, no 1 (Paris: Seuil,
February 1966), pp. 37-49 [translated by Jacqueline Rose as
'Suture (Elements of the Logic of the Signifier)' in Screen, 18: 4
(1978), pp. 24-34. - trans.]. One ought to read along with it Y.
Duroux's article 'Psychologie et logique' appearing in the same
issue (pp. 31-6), which examines in detail the successor function in
2 Cf. A. Badiou, 'Marque et manque: A propos du Zero', in Cahiers
pour l'analyse, no 10 (Paris: Seuil, 1969), pp. 150-73.
3 U'y suis, j'y suis toujours. From Rimbaud's 1872 poem 'Qu'est-ce
pour nous, mon creur, que les nappes de sang' [translated in Col-
lected Poems, ed. and trans. Oliver Bernard (London: Penguin,
1986), pp. 202-3. - trans.].
4 [Miller, 'Suture', p. 40. - trans.]
5 [meconnue. - trans.]
6 [Miller, 'Suture', p. 40. Translation modified. - trans.]
7 [Ibid. - trans.]
8 [See Frege, Foundations, §§ 26-27. - trans.]
9 [Miller, 'Suture', p. 44. - trans.]
10 [Ibid. p. 46. - trans.]
11 [Ibid. p. 47. Translation modified. - trans.]
12 [Ibid. p. 43. - trans.]
13 [I'instance de La Lettre. - trans.]
14 [Ibid. p. 44. - trans.]
15 On the typology of orientations in thought, d. Meditation 27 of
L'P.tre et /'evenement, pp. 311-15 [pp. 281-5 in the English transla-
tion. - trans.].
16 [Miller, 'Suture', p. 40. Translation modified. - trans.]
17 [Ibid. p. 41. - trans.]
18 [Ibid. p. 47. Translation modified. - trans.]
19 ['Matrice', in Ornicar? 4 (1975); translated by Daniel G. Collins in
Lacanian Ink 12 (Fall, 1997): pp. 45-51. - trans.]
20 [Miller, 'Suture,' p. 39. - trans.]
21 [fourmillement: if not for its unfamiliarity, the more direct
etymological equivalent of the psychiatric term formication,
designating a prickling or tingling as of ants crawling over the
skin, might carry less of an inappropriate sense of dynamic
self-organisation than 'swarming': rather than implying any
vital movement, Badiou's fourmillement seems to denote 'our'
phenomenological registration of the icy 'constellations' of Number.
- trans.)
22 For example E. Borel, 'La Philosophie mathematique et I'infini',
Revue du mois, 14 (1912), pp. 219-27.
Chapter 4 Dedekind
1 The reference text for Dedekind's doctrine of number is The Nature
and Meaning of Numbers [see ch. On 1 above - trans.) The first
German edition was published in 1888.
2 [Dedekind, Numbers, § 1. - trans.)
3 [Ibid., §2. - trans.)
4 [Ibid., § § 21-25. - trans.)
5 [Ibid., §§ 26-35. - trans.)
6 [Ibid. §71. - trans.)
7 [Ibid. §73. Dedekind's text has <I> where Badiou uses f - trans.]
8 We might say that Frege is a Leibnizian, Peano a Kantian, and
Cantor a Platonician.
The greatest logician of our times, Kurt G6del, considered that
the three most important philosophers were Plato, Leibniz and
Husserl - this last, if one might say so, holding the place of
The three great questions posed by mathematics were thus:
1 the reality of the pure intelligible, the being of that which math-
ematics thinks (Plato);
2 the development of a well-formed language, the certitude of
inference, the laws of calculation (Leibniz);
3 the constitution of sense, the universality of statements (Kant,
9 [See Dedekind, Numbers, Preface to the first Edn. - trans.)
10 [Dedekind, Numbers, § 64n. - trans.)
11 [Ibid. - trans.)
12 [Ibid., § 66. Dedekind's text has <I> where Badiou has f, and a, b
rather than Sl, S2. - trans.]
13 ~ a - also 'id'. - trans.)
Chapter 5 Peano
The reference text for Pea no is a text published in Latin in 1889,
whose English title is: 'The Principles of Arithmetic'. The English
translation of this text is found in J. van Heijenoort (ed.), From
Frege to Gadel, pp. 83-97.
2 [Ibid., p. 85. - trans.)
3 [Ibid. - trans.)
4 This passage is taken from a letter from Dedekind to Keferstein,
dating from 1890. The English translation can be found in van
Heijenoort (ed.), From Frege to Gadel, pp. 98-103.
5 [Van Heijenoort (ed.), From Frege to Gadel, p. 85. - trans.)
6 [Title of Husserl's 1911 'manifesto'; translated in Q. Lauer (ed.),
Phenomenology and the Crisis of Philosophy (New York: Harper,
1910). - trans.)
7 [Van Heijenoort (ed.), From Frege to Gadel, p. 85. - trans.)
8 [Ibid., p. 85. - trans.)
9 [Ibid., p. 94. - trans.]
10 [Ibid. - trans.)
11 [Ibid. (Axiom 6). - trans.)
12 [froide d'oubli et desuetude, une Constellation: Mallarme, 'Coup de
des', p. 181. - trans.]
13 Regarding these questions, one might read the (purely historical)
chapter 10 of Robinson, A. Non-Standard Analysis (Amsterdam:
North-Holland, revised edn 1974). Robinson recognises that
'Skolem's work on non-standard models of Arithmetic was the great-
est single factor in the creation of Non-Standard Analysis' (p. 278).
For a philosophical commentary on these developments, cf. A.
Badiou, 'Infinitesimal Subversion', in Cahiers pour l'analyse, no 9
(Paris: Seuil, 1968) pp. 118-37.
14 [(a - trans.)
Chapter 6 Cantor:
'Well-Orderedness' and the Ordinals
1 Cantor's clearest articulation of his ordinal conception of numbers
is found in an 1899 letter to Dedekind. See the English translation
of the key passages of this letter in van Heijenoort (ed.), From Frege
to Gadel, pp. 113-17. Cantor demonstrates an exceptional lucidity
as to the philosophically crucial distinction between consistent
multiplicities and inconsistent multiplicities. It is to him, in fact, that
we owe this terminology.
2 On this point, you are naturally referred to the work of Alexander
Chapter 7 Transitive Multiplicities
1 [prenez ensemble - with the intended resonance of ensemble ('set').
- trans.]
2 [decoupe: a 'carving out' or deduction. - trans.]
Chapter 8 Von Neumann Ordinals
1 John von Neumann gave a definition of ordinals independent of the
concept of well-orderedness for the first time in a 1923 German
article, entitled 'On the introduction of transfinite numbers'. This
article is reproduced in English translation in van Heijenoort (ed.),
From Frege to Gadel, pp. 346-54.
The definition of ordinals on the basis of transitive sets seems to
have been taken up again in an article in English published in 1937
by Raphael M. Robinson, entitled 'The theory of classes, a modifica-
tion of von Neumann's system' (journal of Symbolic Logic, no 2,
2 Throughout this book, the ordinals, denoted in current literature by
the Greek letters, will be denoted by the letters Wand w, supple-
mented further on with numerical indices, W h or W3, etc. In general,
W or w designate a variable ordinal (any ordinal whatever). In
particular, we employ the expression 'for every ordinal W'. The
notation with indices is used to designate a particular ordinal, as in
the expression 'take ordinal W, which is the matter of Number N,'.
The subscripts will be used most often to the left of the sign E, to
designate an ordinal which is an element of another, as w, E W
(ordinal WI is an element of ordinal W).
3 [/a Nature (as opposed to nature). - trans.]
4 The Axiom of Foundation, also called the Axiom of Regularity, was
anticipated by Mirimanoff in 1917, and fully clarified by von
Neumann in 1925. To begin with, it was a matter, above all, of
eliminating what Mirimanoff called 'extraordinary sets', that is,
those which are elements of themselves or contain an infinite chain
of the type ... E an+, E an E ••• E az E al E E. It was realised only
later that this axiom enabled a hierarchical presentation of the uni-
verse of sets.
For a historical and conceptual commentary on this axiom, cf.
A Fraenkel, Y. Bar-Hillel and A. Levy, Foundations of Set Theory
(Amsterdam: North-Holland, 2nd edn 1973), pp. 86-102.
For a philosophical commentary, see Meditation 18 of L'P-tre
et /'evenement, pp. 205-11 [pp. 184-90 in the English translation.
- trans.].
5 A good presentation of the fact that belonging (e) orders the
ordinals totally (strict order) - in other words that, given two
different ordinals WI and W
, either WI e W
or W
e WI -
can be found in Shoenfield, J. R., Mathematical Logic (Reading,
MA: Addison-Wesley, 1967), pp. 246-7. This proof is reproduced
and commented upon in L'P-tre et l'evenement in the third section
of Meditation 12, pp. 153-8 [pp. 134-9 in the English translation.
- trans.].
6 [decouper. - trans.]
Chapter 9 Succession and Limit. The Infinite
1 Badiou, A., Manifeste pour la philosoph ie, Paris: Seuil, 1989 [trans-
lated by N. Madarasz as Manifesto for Philosophy (Albany, NY:
State University of New York Press, 1999). - trans.). The circum-
stances and the effects of the philosophy's suture to the poem,
beginning with Nietzsche and Heidegger, are described briefly in
chapter VII, entitled 'The Age of Poets'.
2 [Osip Mandelstam, from his Tristia (1922). Badiou quotes Tatiana
Roy's French translation: vers ces prairies infinies 014 Ie temps
s'arrete. - trans.]. The instant of Presence is beyond all insistence,
all succession. The 'eternal midday' is the trans-temporal limit of
time. Here is the conjoint site of the poem and the sacred.
It is not always in this place, it must be said, that Mandelstams's
poems establish themselves. For in his most powerful poetry he
seeks to think the century, and succeeds in doing so.
Chapter 10 Recurrence, or Induction
1 For the demonstration of the validity of definitions by induction,
you are referred to K. J. Devlin's Fundamentals of Contemporary
Set Theory (NY: Springer-Verlag, 1980), pp. 65-70 ('The principle
of recursion').
Chapter 12 The Concept of Number:
An Evental Nomination
1 To repeat, the basic text for the study of the numbers called 'surre-
als' is Gonshor's Introduction to the Theory of Surreal Numbers
[see ch. 1 n 5 - trans.]. The fact that Gonshor and all current theorists
of these numbers, which I call Numbers, see them as a 'macro-
field' of the reals results in a presentation quite different from my
The inital idea of their creator, Conway, was to define 'surreal'
numbers directly by means of cuts. A number will be defined as a
pair of two sets of numbers, conforming to the conditions of the
cut (every number in the set 'to the right' in the pair is smaller than
every number in the set 'to the left'). The double circularity of this
definition obviously must be questioned (number is defined on the
basis of number, and inequality between numhers is mentioned
without having been properly defined). The operation that serves to
undo this circle is obviously transfinite induction, which makes
ordinals appear inevitably on the scene. In fact, Conway presents
Numbers on the basis of their canonical representation - that is, in
my language, their 'structural' character: they are defined on the
basis of their sub-Numbers. D. E. Knuth's book Surreal Numbers
(Reading, MA: Addison-Wesley, 1974) gives a 'pedagogical' version
of Conway's presentation in the form of a dialogue. It seeks to rec-
reate the mentality of a 'researcher' into the matter, but in fact
becomes quite convoluted, since in its exposition the employment
of the ordinal series is not made explicit. Besides this, it re-
establishes, to my mind to the detriment of the real 'genius' of the
invention of Numbers, a creationist and progressive logic (first
'creating' zero, then 1 and -1, etc.)
Gonshor starts from a literal 'coding', whereas, in my quest for
the concept and its philosophical deployment, I join a set-theoretical
lineage. Technically speaking, Gonshor generalises the development
in base 2 of the real numbers. A real number can he presented as
an infinite denumerable series of signs 1 and O. Gonshor's idea is to
consider such series of any ordinal length whatsoever, rather than
limiting them to denumerahle series. He then begins with two signs
+ and -, and calls 'surreal number of length W' a series of such signs
indexed to the elements of the ordinal W. The index ordinals affected
by the sign + correspond to the elements of what I call the form of
the Number, and the index ordinals affected by the sign -, to the
elements of the residue. The ordinal 'length' corresponds to what I
call the matter of the Number.
As an example: the Number which I write (4,(0,3)), whose matter
is 4 and whose form contains the elements 0 and 3, is written by
Gunshor as follows: + - - +.
Now, of course surreal numbers and Numbers are 'the same
thing'. But we might say that Gonshor treats them as inscriptions,
or markings, after the manner of Frege and of Peano's arithmetic.
The inspiration here is ideographic. Whereas I approach them from
the point of view of their multiple-being, in the Cantorian spirit,
my inspiration being ontological, or Platonic.
In fact even the technical development ends up being quite dif-
ferent, although the results can always be translated from one
version to the other. For example, it is not insignificant that
Gonshor, who, with the signs + and -, is unable to denote an
occurence of the void, must invoke an 'empty series' of signs,
where I would write (0,0). The conceptual advantage of the
ontological approach to Number is that it allows one to dispense
with all additional literalisation, with every heterogeneous sign,
in favour of the two fundamental set-theoretical relations of
belonging E and inclusion c. This doubtless explains why for
Gonshor the theory of surreal numbers is a sort of specialist
technique, whereas for me it is a wholly natural extension of the
ontological vocation of set theory to the concept of Number.
2 Gonshor, Introduction, p. 43.
3 [decoupe. - trans.]
Chapter 13 Difference and Order of Numbers
1 Throughout this book, I call a relation (most often one of order)
'total' when two different basic terms of the relation are always
bound by this relation. Thus I would say that the relation E is total
in the ordinals or that the relation E is total in the Numbers.
Sometimes a relation is called 'total' which is also reflexive,
binding each term to itself. This is the case, for example, with the
relation:::;; (less than or equal to) for the natural whole numbers.
Limiting oneself to my definition, which only demands the rela-
tion between different terms, and excludes the relation of self with
self (an irreflexive relation, then), is more convenient in the case we
are dealing with. Where we speak of an order-relation, we mean to
say that its axioms are those of strict order.
2 For Gonshor, order is easily presented as lexicographical, since surreal
numbers are introduced as series of signs + and - (d. ch. 12n 1).
3 On this point, d. Miller, J.-c. Libertes, Lettre, Matiere (Paris: Con-
ference du Perroquet, June 1985).
4 [Paul Celan, from Zeitgehoft (1976). [Badiou's reference is M.
Broda's translation (Paris: Clivages, 1985): chiquenaude / dans
l'abime, dans Ies / carnets de gribouillages / Ie monde se met a bruire,
il n'en tient / qu'a toi. - trans.]
Chapter 14 The Concept of Sub-Number
1 Category theory is an attempt to reformulate all of mathematics
within a structural, non-set-theoretical framework whose starting
point is 'objects', which are 'types of structures'; and 'arrows',
which are transformations, or morphisms, between structures. The
concept of substructure can be understood in terms of that of sub-
object. A 'sub-object' is in fact an equivalence class for certain
arrows. Cf. for example J. L. Bell's book Toposes and Local Set
Theories (Oxford: Clarendon Press, 1988), in particular the argu-
ments of pp. 49-58.
2 [sectionne. - trans.)
3 ['a la matiere pres': In mathematics, 'a la x pres' - English equivalent
'up to x' - indicates that abstraction is to be made from a certain
class of objects, which for the purposes of a particular statement or
definition are to be regarded as a single entity. Thus a certain propo-
sition can be said to be true, or a property to be satisfied, 'up to
isomorphism', 'up to rotation', 'up to translation', and so on. In the
present context, the 'cut' between the high and low sets uniquely
defines a Number, so long as we regard all possible configurations
of 'matter' as being subsumed under the aspect of the unique minimal
case. In other words, the definition of the cut must be supplemented
by the principle of minimality. - trans.]
4 [encadrement: an interval in the mathematical sense, as in 'interval
around a real number'. - trans.]
Chapter 15 Cuts: The Fundamental Theorem
1 The problem of the cardinality of the set of parts of an infinite set is
a central problem for set theory after Cantor. The 'minimal' hypoth-
esis, which says that this cardinality is the smallest cardinal larger
than that of the initial set - the cardinal successor of the one which
measures the quantity of that set - is the famous 'continuum hypoth-
esis', denoted by CH in the English literature on the subject.
Following P. H. Cohen's work, we know that the continuum
hypothesis is undecidable on the basis of the classical axioms of the
theory. It can be affirmed or denied without any contradiction being
A particularly lucid text on this problem is K. Godel's 'What is
Cantor's continuum problem?'. The English text has often been
republished since its first appearance in 1947; for example in P.
Benacerraf and H. Putman (eds), Philosophy of Mathematics (Cam-
bridge: Cambridge University Press, 2nd edn, 1983) pp. 470-86.
2 [dans ces parages du vague OU toute rea lite se dissout: Mallarme,
'Coup de des', p. 181. - trans.]
3 [Ibid. - trans.]
4 The concept of the cut, and the way in which it specifies the relation
between punctual intervention and the continuum of situations,
traverses all the procedures of truth. Its occurrence can be remarked
in the politics of rupture (revoutionary politics), in the artistic theme
of novelty or of modernity, in the scientific theme of crises and
refoundations, or in the amorous figure of separation. Every fidelity
is also the process of a cut.
S Dedekind's fundamental text on the idea of the cut, dating
from 1872, is 'Continuity and Irrational Numbers' [translated in
Dedekind, Numbers, pp. 1-24. - trans.].
6 The exposition in Gonshor, Introduction, begins with the demon-
stration of the fundamental theorem. His style is very different: both
because, as I have already mentioned, Gonshor adopts a line which
is oriented more towards calculation than towards set theory; and
because he is not content with a proof of existence, but intends to
determine exactly the Number that is cut (what is called a 'construc-
tive' proof). This concern for determination entails the examination
of a great many cases.
7 We will see in chapter 16 that the upper bound of a set L, being of
the form (WI, Wd, is an ordinal. This is a striking result.
8 The lower bound of a set H is in fact the negative of an ordinal, a
Number -(W). Cf. ch. 16.
9 For rule 2, the reasoning is exactly symmetrical to that which vali-
dates rule 1. Let us take rule 3: we have Id.(W, Nb), and W is in
the form of Nb. I put W in the form of Ni. Am I not risking making
it so that Ni becomes thus as large as a Number of A? Take Na to
be this supposed Number. W must be the discriminant of Ni and
of Na, which is to say that it is also the discriminant of Na and Nb.
Now W is in the form of Nb, one must therefore have Na < Nb,
which is not allowed.
The same approach can be applied for rule 4.
Chapter 16 The Numberless Enchantment of the
Place of Number
1 As we have indicated in notes 7 and 8 of the preceding chapter, a
very interesting 'topological' characteristic of the positive and nega-
tive ordinals is that every set of Numbers has an ordinal as its upper
bound, and the negative of an ordinal as its lower bound. This can
be explained easily enough, since every ordinal is the cut of itself
and the void, and every negative of an ordinal, a cut of the void
and itself.
2 Gonshor, Introduction, p. 32.
3 [encadrement. - trans.]
4 The principle of the isomorphism of orders - that is, of the fact that,
if N is a Number of finite matter and RA(N) = r, then Nl < N
RA(Nd < RA(N
) - is simple enough (note that < is to the left of
the implication in the order-relation in Numbers, to the right of the
relation of ordinary order in the rational numbers). The result is
that, in the decomposition of N in the form 1 + 1 + ... - t + etc.,
what is added 'at the end' decreases very rapidly. This is a quite
simple, normal algebraic calculation.
5 See Gonshor, Introduction, pp. 30-1.
6 One might object at this point that our Numbers do not authorise
the representation either of complex numbers or of quaternions,
upon which physics relies to a considerable extent.
But are complex numbers and quaternions numbers? I think it
can be reasonably maintained that, from the moment we take leave
of all 'linearity' when we abandon dimension 1, we are dealing
with constructions based on Numbers rather than with Numbers
per se.
Basically, the innermost essence of complex numbers is geometri-
cal, it is the 'complex plane' which delivers the truth of these
'numbers'. Around the complex numbers is organised the profound
link between pure algebra (the extension of fields) and the ontologi-
cal scheme of space as topological concept. I am tempted to call
complex numbers operators, operators whose function in thought
is to articulate algebra and topology. Hence the simultaneously
combinatorial (a complex number being a pair of real numbers) and
geometrical character of these 'numbers'. They are in fact numbers
which do not number, but suggest schemes of representation and
inscription which are already, in effect, something very close to a
conceptual 'physics'.
Moreover, it seems to me unreasonable to speak of 'numbers'
when it is not even possible, in terms of the operational field con-
sidered, to say that one 'number' is larger or smaller than another.
In short: a field of numbers must in my view be an ordered field,
which neither complex numbers nor quaternions are.
Finally, I restrict the concept of Number, in so far as it is thought
of as a form of being, to that which can be deployed according to
the intuition of a line. This is made clear by the decisive part played
in the definition of the being of Number by that fundamental 'line
of being' constituted by the ordinals.
7 See Robinson, Non-Standard Analysis [see ch. Sn 13. - trans.].
Chapter 17 Natural Interlude
1 On the mathematical personality Ramanujan, see the great number-
theorist G. H. Hardy's autobiographical A Mathematician's Apology
[Cambridge: Cambridge University Press, 1940. - trans.].
2 On the set-theoretical reduction of relations and functions to the
pure multiple, and for an ontological discussion of this point, see
L'P.tre et l'evenement, Appendix 2, 'A relation, or a function, is
solely a pure multiple', pp. 483-6 [pp. 443-7 in the English transla-
tion. - trans.].
3 [La nature a lieu, on n'y a;outera pas: Mallarme, 'De la musique et
des lettres', in H. Mondor and G. Jean-Aubry (eds), Oeuvres com-
pletes (Paris: Ple"iade, 1945) pp. 642-57. Translated as 'Music and
Literature', in B. Cook (trans.), Mallarme: Selected Prose Poems,
Essays and Letters (Baltimore: Johns Hopkins Press, 1956),
pp. 43-56 (translation modified). - trans.].
Chapter 18 Algebra of Numbers
1 It is equally true that every set of Numbers whose matter is lower
than or equal to a given infinite cardinal is a commutative field. In
this regard, Gonshor is right to say that the study of the field of
Numbers of finite matter or equal to ro ('of countable length' [see
Gonshor, Introduction, p. 103. - trans.]) would be most worth-
while. This field allows real Numbers as a subset, but it also contains
infinitesimals and cuts of cuts. It would be possible to develop a
wholly original analysis here.
2 Gonshor, Introduction, Ch. 3, 'The Basic Operations'.
3 Take two ordinals WI and W
, where (((W
») = W. If WI is
maximal in the couple, every couple (WhW4) where W4 E W
smaller than the couple (W.,W
) in the order of couples (see 17.6),
because they have the same Max (which is WI! and the same first
term (which is also WI!, but the second term of the couple (W
is smaller than the second term of (W
). So, (((W
»), since (is an isomorphism of order between couples of
ordinals and ordinals.
If it is W
that is maximal, the same conclusion follows, since the
Max of (W
W4) is lower.
Similar verifications can be made for any such case.
4 The induction in question consists of proving simultaneously:
• that, if N2 < N3, then NI + N2 < NI + N3 (compatibility of order
and additive structure);
• that the Numbers of set L are all smaller than the Numbers of
set H.
5 To be really meticulous, we must take into account the case where
L does not contain any positive Numbers, but does contain O. In
this case, 0 is the internal maximum of L. One can take it as 0
'alone', or identify L with the set (0). The reasoning is then much
6 The reader might be perturbed by the constant amphibolies of nota-
tion (the sign < used in one place for the order of Numbers, else-
where for that of this or that particular type of number, etc.) In fact,
mathematicians (who say in such a case 'that there is no possible
equivocity') express through such amphibolies their natural ten-
dency to identify purely and simply, and therefore to name identi-
cally, relations and operations which are defined with isomorphic
structures. How else could Category Theory have arisen, taking as
its 'primitives' not multiplicities, but 'morphisms', or arrows, des-
ignating 'correspondences' between structural 'objects'?
Chapter 19 In Conclusion:
From Number to Trans-Being
1 [action restreinte: Mallarme, 'Action restreinte', in Oeuvres com-
pletes (see ch. 17n 2), pp. 369-73.]
2 [Como urn deus, nao arrumei nem uma coisa new outra: from
Alvaro de Campos aka Fernando Pessoa's 1929 poem 'Reticencias'.
See F. Pessoa, edited by M. A. D. Galhoz, Obra Poetica (Rio de
Janeiro: Aguilar, 1960). This is a variant of his 'Quasi' (cf. Vol. II
of the Edirao Critica (lmprensa Nacional- Casa da Moeda, 1990),
p. 215), where we read Como um deus, nao arrumei nem a verdade
nem a vida ('Like a god, I arranged neither truth nor life'). Badiou
quotes A. Guibert's French translation: Tel un dieu, je n'ai mis de
l'ordre ni dans l'un ni dans l'autre. - trans.]
addition, 198-9
and inductive definition, 95, 96,
see also calculation
adherence, 79, 87
algebra, 142, 196, 197, 198-208
Numbers of finite matter and
dyadic rationals, 173-4
as operational dimension of
Number, 111
and substructure, 132, 133
algebraic orientation, 10, 12,48
All, 14,87,88-9, 139, 144, 193
alogos and nomination, 106
Althusser, Louis, 3
analytic geometry, 11
Aristotle, 109
arithmetic, Peano's signs, 49-50,
arithmetic/geometry opposition,
10-11, 12, 141
art, 3, 111
as truth procedure, 57
see also poetry
Axiom of the Empty Set, 44, 57
Axiom of Foundation, 70-2
Axiom of Infinity, 57, 82, 94
Axiom of Separation, 21, 43
axiomatic approach to number, 9,
and decision, 212-13
Peano's axioms, 8, 11, 12,46-51
base-element of number
(Dedekind), 33, 34
beatitude, intellectual state of, 144,
Being and number, 9, 12, 25, 57-8,
and operational properties, 200
and ordinals, 82, 83
trans-being, 214
zero, 157-8
see also existence; ontology of
being of the One, 7, 8, 13, 55
belonging, 61-6, 68, 69-70, 71,
and difference between numbers,
and sets of ordinals, 112, 113-14
biunivocal correspondence
couples of ordinals, 193-5, 196,
biunivocal correspondence (cont')
and Dedekind, 31, 32-3, 34,
and Frege, 13, 16-17
Bolzano, Bernard, 10, 13
Borel, Emile, 30
Bourbaki, N., 10
Burali-Forti paradox, 55-6
bureaucratisation of knowledge, 2
as essence of number, 8, 180
and truth, 26, 27
see also addition; algebra
canonical presentation of the cut,
157, 165
Cantor, Georg, 1, 3-4, 8, 12, 13,
52-8,69, 73,212
excess of parts over elements
theorem, 63, 65, 66
transfinite numbers, 106
cardinals, 10, 12, 13, 226n
Frege, 11, 17, 31
and zero, 34
Carnap, Rudolf, 48
category theory, 131
Cauchy, Augustin Louis, Baron,
Celan, Paul (poem), 130
ciphers, 95
Cohen, P.H., 139
commutative field, 199, 204
complete induction, 86
completion of Number and the cut,
complex numbers, 10, 11, 12, 228n
and quaternions, 228n
concept of number, 9-15, 58, 69,
complex numbers and
quaternions, 228n
Dedekind, 34-5
definition of Number, 101, 102-
12, 133
definition of operation on
Numbers, 201
difference and order of numbers,
Frege, 16-23, 55
and multiple-being, 107
Number as form of Being, 211
sub-number concept, 131-8
and terminology, 105-7
unification of number and
operation, 200
constructivist orientation of
thought, 27, 30
continuous and discrete, dialectic
continuum hypothesis, 10, 226n
and the cut, 141-2
Conway, J.H., 9, 107
counting, 9, 55, 58, 184
algebra, 198-208
countable and definition of
number, 109, 111
and empty sets, 64
and Peano's axioms, 51
and political value, 1-2
and society, 3
couples of ordinals and operation,
184-96, 201
cultural representation, 2-3
cuts and cutting, 139-55
addition, 201-8
construction of a cut, 147-53,
cut of cuts, 179-80, 229n
Dedekind's theory, 11, 106,174-
5, 176-7, 178
and dyadic rationals, 174-5,
and fundamental theorem of
ontology of Number, 143-55,
156, 201, 203
and operation, 201
principle of the cut, 143
sub-Numbers, 136, 138
surreal numbers, 224n
decision, 212-13
decomposition and structure, 131,
Dedekind, R., 1,3-4, 8, 12, 13-14,
the cut, 11, 106,141,142,174-
5, 176-7, 178
letter to Keferstein, 47-8
and Pea no, 46-7, 50
system of the infinite, 32, 35-44,
deduction, 47, 212
definition of addition, 96
definition of Number, 101, 102-12,
Deleuze, Gilles, 49
denominators, 167, 169
density of order
and the cut, 140, 141-2, 142-3,
155, 156
infinitesimal numbers, 179
see also swarming of numbers
Descartes, Rene, Cogito, 38-9,
difference between matter and form
of Number, 111, 115-30
and order over numbers, 119-
30, 156
discrete magnitude, 141
discriminant, 117, 156
discrimination between numbers,
and cutting, 148, 150-2, 154,
156, 180
and sub-Number concept, 135-7
Zero, 158-9
disorder and restrained action, 214
dissemination, 79, 80-1, 85
dyadic positive rational numbers,
economy, 3
'economy of number', 48
elements of the elements of a set,
empty extensions, 18-19
empty sets, 22, 55, 63-4, 83, 113,
and difference between numbers,
and succession, 77, 84-5, 95
and von Neumann ordinals,
67-8,69, 71
empty sets axiom, 44, 57
equals sign and Peano, 49-50
equinumerate concepts, 16-17, 18,
Euclid, Elements, 7, 36, 106
evental sites, 107
evental trans-being, 214
events and truth process, 27, 155,
Evil, 160-1
excess of parts over elements, 63,
65, 66, 139, 140
and fundamental theorem of
ontology of Number, 146-53
of infinite, 37-44,47-8, 55, 56,
of limit ordinals, 82
of zero, 22-3,56-7
see also Being and number;
ontology of Number
extension of a concept, 16-17
Russell's paradox, 20, 21
zero, 18,25
Dedekind and Frege, 14, 30, 44,
Dedekind's definition, 36
finite group theory, 131
finite matter see Numbers of finite
finite number, 32, 45, 97, 166
first limit ordinal (<0), 94-5
'first modernity' of thinking of
number, 13
form of Number, 102, 103-5, 108,
110, 156
and the cut, 147-53, 179, 180
and difference, 115-30
and ordinals, 114
and residue, 111, 112-13, 165
and sub-Number concept,
form of Number (cont')
and symmetrical counterpart,
161-3, 165,174
formalist approach to number, 8, 9,
11, 14
fractions and dyadic rational
numbers, 167, 169, 171
Frege, Gottlob, 1, 3-4, 8, 13-14,
16-30, 44, 212
'cardinal number' concept, 11,
17, 31
and Dedekind, 32, 35, 40
Miller's contemporary usage,
functions, Dedekind, 32-3
fundamental theorem of the
ontology of Number, 143-55
and operation, 201, 203
generic orientation of thought,
analytic geometry, 11
arithmetic/geometry opposition,
10-11, 12, 141
and Greek thought, 10
Godel, Kurt, 8, 10, 139, 220n
Gonshor, Harry, 9, 107, 170-1,
224-5n, 227n, 229n
Greek numbers, 10,93,97
crisis of number, 106
Greek thinkers and number, 7, 10,
Hegel, G.W.F., 19,95, 141, 162
Heidegger, Martin, 81
high sub-Numbers, 135-8, 145,
157, 160, 165, 166, 176,
Hilbert, David, 8
history, 2
history of mathematics, 212
homogeneity and multiplicity of
ordinals, 68-9, 72, 80
human sciences, 2
Husserl, Edmund, 220n
idea of an idea, Dedekind and
Spinoza, 38, 39,40-1
ideals theory, 132
Imbert, Claude, 217n
immensity of Numbers, 102,
107-8, 112,177-8,200,211,
inclusion and transitive sets, 61-6,
68, 113-14
inconsistent multiplicities, 89, 144,
193, 199,212
Pea no, 47
reasoning by recurrence, 86-92,
95, 96, 169
see also inductive definition;
transfinite induction
inductive definition, 89-92, 96-7,
and addition, 95, 96,201-8
and couples of ordinals, 193-5,
infinite, 7, 13-15,212
Cantor's introduction of 0), 53-4,
Dedekind and Frege, 14
Dedekind's definition, 36
Dedekind's system of number,
32, 35-44, 47, 55, 56, 73,
85-6, 93
and natural whole numbers, 93, 97
and Peano's axiom, 47, 51, 57
succession and limit, 77-82, 84,
85-6, 94, 95, 188-96
infinitesimal numbers, 108, 109,
intervallic Number, 148, 152
intuitionist approach to number,
29, 30, 162
irrational numbers
Dedekind's cut, 141
as designation, 106
irreflexivity, 121, 123-4
and algebra, 208
and dyadic rational numbers,
169, 173-4
field of the rea Is, 200
and well-ordered sets, 54-5, 196
iteration of number, 29-30
Kant, Immanuel, 35, 141
Knuth, D.E., 9, 224n
Lacan, Jacques
signifier and the letter, 130
theory of the subject, 24, 25, 28,
lack and function of zero, 26, 28,
and inductive definition, 89
and Pea no, 46, 48-9
terminology of Number, 105-7
law of Capital, 213, 214
Leibniz, Gottfried Wilhelm, 26-7,
30, 35
Principle of Identity, 18-19,25
lexicographical order, 130
limit ordinals, 77-8, 78-9, 80, 81-
2, 84, 85,211
couples of ordinals, 188-96
and inductive definition, 90, 92
and maximal element, 113
and natural whole numbers,
and reasoning by recurrence, 87,
and whole ordinal part of a
Number, 172
'linguistic turn' in philosophy, 48
logic of the signifier, 24-5, 29, 30
logicist approach to number, 8, 9,
11, 16, 18-19,22
and ordinals, 52
Peano's axioms, 46-51
see also Frege
love, as truth procedure, 57
love and the negative, 161
low sub-Numbers, 135-8, 157,
160, 165, 166, 176, 202
lower bound of a set of Numbers,
Lyotard, ].-F., 48
Mallarme, Stephane, 13, 16, 196,
Mandelstam, Osip, 223n
Marx, Karl, 3
mathematics and Number, 212,
matter of Number, 102, 103-5,
108, 110, 111, 113, 156
and the cut, 147-53
and difference, 115-30
and sub-Number concept,
see also Numhers of finite matter
maximal elements and ordinals, 76,
81, 113
maximal ordinal of a couple,
medicine, 2
Michel, Natacha, 13
Miller, Jacques-Alain, 24-30, 39-
minimal element of ordinals, 113,
117, 186-8
minimal matter and cutting, 145-6,
153-5, 157, 176
minimality see principle of
Mirimanoff, D., Axiom of
Foundation, 222-3n
modern analysis, 13, 56, 212
Modernity, 65
and ontology of mathematics,
and ordinals, 108, 109
procession from the One, 7
pure multiple, 8, 44, 58, 65, 83,
101, 185
multiple-being and concept of
number, 107
multiplication, 199
and inductive definition, 95,
96-7, 207
multiplicities, 61-97
Dedekind's theory, 31-45, 56
inconsistent multiplicities, 89,
144, 193, 199,212
multiplicities (cont')
natural multiplicity and ordinals,
68-70,83,84,88-9, 166,211
transitive multiplicities, 61-6
von Neumann ordinals, 67-72
'na'ive' theory of sets, 31-45
natural multiplicity and ordinals,
68-70, 83, 84, 88-9, 166, 211
natural whole numbers, 10, 12,
93-7, 157
addition, 207-8
and Dedekind's cut, 11
definition, 94, 95
and definition of Numher,
108-9, 110
'Peano's axioms', 8, 11, 12,
46-51, 88
positive and negative numhers,
and succession, 77, 84, 95
Nature and number, 69-70, 83, 92,
109, 130, 183-97
negative numbers, 12, 157, 158-61
addition, 205-6, 208
natural whole numbers, 166-7
and ordinals, 165
symmetric counterparts, 161-3,
Negativity, 160-1
Neoplatonism, 7
Neumann, John von see von
neutralisation and cutting, 147,
o as neutral element for addition,
Nietzsche, F.W., 49, 65
nominalist orientation of thought,
27, 175
nomination and signification,
106-7, 108, 175
non-reflexive relations, 121, 123-4,
non-self-identity and zero, 25-30
'not identical to itself' extension
and zero, 18-21, 25-30
definition of Number, 101,
102-12, 133, 228n
and Nature, 69-70
of Numhers, 139, 140
Peano's signs, 49-51
and structure, 131-8
and thought, 10, 88-9, 92
see also concept of number
number theory, 11, 132
and Greek thought, 10
immensity of Numbers, 102,
107-8, 112, 177-8, 200, 211,
Numbers of finite matter, 229n
addition, 207
and dyadic rational numbers,
numerators, 167
numericality of Number, 120, 121,
and symmetricisation, 165-6
objects, 211
category theory and substructure,
Dedekind and infinite, 38-43
Frege's doctrine of, 19,20
mathematical ontology and
multiples, 131-2
and 'not identical to itself'
extension, 25-30
One, the, 7-8, 10, 13-15,55,85
Dedekind and Frege, 14,44
Frege and concept of One, 18
and Peano's axioms, 50
and trans-being, 214
ontology of mathematics, 131-2,
ontology of Number, 8-9, 58, 101-
80, 157-8,211-14
and algebra, 200
and the cut, 175
and dyadic rationals, 174
fundamental theorem of, 143-
meditation on negativity, 160-1
and number as concept, 19,21-
and order of numbers, 121-2
ordinals, 68
pairs of ordinals and succession,
and surreal number theory, 224n,
and transitive sets, 65-6
open sets, 142
operational dimension of Number,
9, 101, 111, 183-208,211
algebra, 196, 197, 198-208
definition of operations on
Numbers, 201
fictions and dyadic rationals,
Numbers of finite matter and
dyadic rationals, 173-4
and whole Numbers, 166-7
see also calculation; counting;
order of Numbers, 12, 101, 111,
119-30, 156,211
definition, 119
density of order, 140, 141-2,
142-3, 155, 156, 179
dyadic rational numbers, 167-8
see also 'well-orderedness' and
ordered pairs of ordinals, 184-96
ordinals, 10, 12,67-97, 156, 157
Cantor's theory of well-
ordered ness, 52-8, 61, 68,
188, 194, 196
couples of ordinals, 184-96, 201
and cuts, 139-55
and Dedekind, 31, 34, 44
and definition of Number, 102-
5, 108-12
difference of numbers, 115-20,
and Frege, 11
and infinity, 44, 85-6, 93, 94
natural whole numbers, 93-7
as Numbers, 163-6
operational ability, 184-97
and order of numbers, 120-30,
156, 164
properties of, 86-92, 94-5, 113,
reasoning by recurrence, 86-92,
95,96, 169
sets of ordinals, 112-14, 139
sub-Numbers as, 165
and succession, 73-82, 84-5, 95,
and surreal numbers, 107
triplets of ordinals, 196
von Neumann ordinals, 67-72
whole ordinal part of a Number,
'outside matter' position, 120,
particular predicates, 8
partitioned Number, 133-4, 201
parts of ordinals, 102, 103-5, 108,
109, 112, 113-14
Pascal, Blaise, 32
Peano, Giuseppe, 1, 3-4, 9, 13,
axioms, 8, 11, 12,46-51, 88
Pessoa, Fernando (poem), 214
philosophy and language, 48-9
place of number, 93, 156-80
Dedekind, 35, 36-7, 39, 42
and succession, 95
Plato, 35, 49, 64, 103
'platonising' approach to number,
poem, 'suture' of philosophy, 81
poetry and the negative, 161
Political Science, 2
governed by numbers, 1-2
and the negative, 161
as truth procedure, 57
revolution vs reform, 79
positive Numbers, 158-60
addition, 205-8
dyadic positive rational numbers,
positive Numbers (wnt')
infinitesimal numbers, 178-9,
natural whole numbers, 166-7
and ordinals, 165
symmetric counterparts, 161-3
and whole ordinal part of a
Number, 170-3
Presence, 56, 57
and belonging, 65, 66, 68, 71
and structure, 131, 133
principle of the cut, 143
principle of identification, 143
Principle of Identity, 18-19,25
Principle of Indiscernibles, 27, 30
principle of minimality, 72, 88, 94-
5, 113, 117, 186-7
and cutting, 143, 144, 146, 153-5
Principle of Non-Contradiction, 27
principle of recurrence, 12, 47
see also reasoning by recurrence
Principle of Sufficient Reason, 19
procedure of neutralisation, 147,
progression of number, 29-30
properties of ordinals, 86-92, 94-
5, 113, 134
and whole ordinal part of a
Number, 170-2
property of succeeding, 75-6
proportions and 'irrational
numbers', 106
pure multiple, 8, 44, 58, 65, 83,
101, 185
quasi-continuity of rationals, 142
quaternions, 10, 228n
Ramanujan, Srinivasa, 183-4
rational numbers, 10, 12, 109,
157, 199
and Dedekind's cut, 141-2, 155,
174-5, 176-7
dyadic positive rational numbers,
and Frege, 11
real numbers, 10, 12, 107, 109,
and algebra, 199
and Dedekind's cut, 11, 142,
174-5, 176-7, 179-80
definition of, 176, 199-200
and dyadic rationals, 174-7
and Frege, 11
infinitesimal numbers, 178-9,
positive real numbers and cut of
cuts, 179-80, 229n
and Peano's axioms, 12,47
reasoning by, 86-92, 95, 96, 169
see also inductive definition
reign of number, 213, 214
relative whole numbers, 10
repetition and number, 29-30
representation and inclusion, 65,
66, 68
residue of the Number, 102, 104,
105, 111, 112-13, 156
and the cut, 145, 147-53, 180
and difference, 115-30
and dyadic rational numbers,
168, 169, 172
and sub-Number concept, 134
and symmetrical counterpart,
161-3, 165, 174
upper bound of Number, 145
and zero, 159, 160-1
restrained action, 213, 214
ring theory, 132
Robinson, A., 50, 178
Russell, Bertrand, 8,212
Frege and Russell's paradox, 20,
salva lleritate, 26, 27
Sartre, Jean-Paul, 39
science and the negative, 161
sciences and numbers, 2
'second modernity' of thinking of
number, 13-15
section of Number, 108-9, 110,
serial numericality, 88, 121-2
series of numbers, 7
set of concepts, 17, 20
set of parts, 91
set-theoretical approach to number,
8-9, 11,22
set theory
axiom of foundation, 70-2
Dedekind's 'na"ive' theory of sets,
set of all sets, 42-3
and surreal numbers, 224n, 225n
sets of ordinals, 112-14, 139
sign/signifier and the letter, 130
sign/signifier and number, 24-5, 29,
Gonshor and surreal numbers,
nomination and signification,
106-7,108, 175
Pea no's axioms, 48, 49-51
similar transformation, Dedekind,
32, 34
singletons, 64-5, 71, 95
Skolem, T., 50
society and number, 3
sociology, 2
Spinoza, Baruch, 38-9, 40-1
'square circle' concepts, 18, 19
square numbers and biunivocal
correspondence, 36
square root, 199
statistics, 2
structure and substructure, 131-8,
subject, Lacanian theory, 24, 25,
subjectivity of number
Dedekind, 39-40
Miller's challenge to Frege,
sub-number concept, 131-8, 157
and cutting, 145, 147, 148, 154,
and dyadic rationals, 176
and natural whole numbers, 166
and operation, 201, 202
sub-Numbers as ordinals, 165
sub-Numbers as positive, 160
sub-objects and category theory,
subsets of ordinals, 102, 103-5
substitutability, 26
substructure, 131-8, 147
succession, 30, 73-82, 84, 85, 95,
142, 188-96,211
successor ordinals, 76-8, 79, 81,
couples of ordinals, 188-96
Dedekind, 34, 46-7
and discriminants, 117-18, 119
and inductive definition, 89-90,
Peano's signs, 49, 50
and whole ordinal part of a
Number, 172
surreal numbers theory, 9, 13,
use of 'surreal', 107, 108
swarming of numbers, 30, 112,
139, 140, 200
and fundamental theorem of the
ontology of Number, 143-55
symmetric counterpart of a
Number, 161-3, 165-6, 174
symmetricisation of addition, 12
symmetricisation of multiplication,
system of number, Dedekind, 32-8
terminology of Number, 105-7
thought and number, 1-4,57-8,
the cut, 155
and Dedekind's infinite systems,
Frege and concept of number,
Greek thinkers, 7, 10,57
and language, 48-9
topological orientation, 10, 12, 142
topology of difference, 120
total relations, 121, 122-3, 156
trans-being, 214
transfinite induction, 19, 86, 89,
and couples of ordinals, 193,
and operation, 201-8
and surreal numbers, 224n
transfinite numbers, 106
transformation, Dedekind, 32-3,
transitive sets
and inclusion, 61-6, 68, 113-14
and succession, 74-5
and von Neumann ordinals,
transitivity, 12
difference and order of numbers,
121, 125-30, 156
triplets of ordinals, 196
and calculation, 26, 27
and Number, 213-14
truth process and event, 27
truth-case of concepts, 16, 25
ultra-One, 214
unconscious, 41
unicity and fundamental theorem
of ontology of Number, 145,
union of a set, 79-80, 81, 91, 102
uniqueness and the cut, 143,
144-6, 155, 157
unity, 7, 8
Peano's signs, 49, 51
upper bound of ordinals, 113, 189,
of a set of Numbers, 144-5
value and Number, 213-14
Venn diagrams, 105
void, 157-8,212
and Evil, 160-1
and inductive definition, 89, 91
singletons of, 64-5, 71, 95
and succession, 77, 84-5, 95
see also empty sets; zero
von Neumann, John, 8, 12
von Neumann ordinals, 67-72,
and succession, 73, 74-5
'well-ordered ness' and ordinals,
52-8,61,68, 188, 194, 196
whole numbers see natural whole
numbers; relative whole
whole ordinal part of a Number,
Wittgenstein, Ludwig, 212
Zermelo, E.F.F., 8, 22
Axiom of Separation, 20-1, 43
zero, 7-8, 13-15
and axiom of the empty sets,
and being, 25, 56-7
and belonging and inclusion,
Dedekind and Frege on, 14, 40,
discriminant, 158-9
existence of, 22-3, 56-7
Frege and concept of zero,
18-19,22, 25-30, 40, 55
and infinitesimal numbers,
lack and function of zero, 26,
28, 34
as neutral element for addition,
as Number, 157-8
ordinals and well-ordered sets,
and Peano's axioms, 50, 51
and positive and negative
Numbers, 158-61
see also empty sets; void