## 1. Infinity in philosophy: some historical remarks



In Greek, 'to apeiron' means 'the infinite':
'a' denotes privation and 'peras' the notion
of 'limit' or 'bound'. Etymologically, the
English word 'infinite' comes from the Latin word
'infinitas': 'in' = 'not' and
'finis' = 'end', 'boundary',
'limit', 'termination', or 'determining
factor'. In contemporary English, there is a range of uses of
the word 'infinite':


1. In a loose or hyperbolic sense, 'infinite' means
'indefinitely or exceedingly great', 'exceeding
measurement or calculation', 'immense', or
'vast'.
2. In a strict but non-mathematical sense that reflects its
etymological history, 'infinite' means 'having no
limit or end', 'boundless', 'unlimited',
'endless', 'immeasurably great in extent (or
duration, or some other respect)'. This strict, non-mathematical
sense is often applied to God and divine attributes, and to space,
time and the universe.
3. There is also a strict, mathematical sense, according to which
'infinite' quantities or magnitudes are those that are
measurable but that have no finite measure; and 'infinite'
lines or surfaces or volumes are measurable lines or surfaces or
volumes that have no finite measure.



Related to the distinction between meanings (2) and (3) is a
distinction between metaphysical and mathematical meanings of
infinity. This has been usefully employed in some of the most
encompassing accounts of infinity, such as Moore (1990/2019; for
another recent treatment that includes extensive discussion of the
history of infinity see Zellini (2005)). Moore sees the metaphysical
notion as bound up with the notions of 'totality',
'absoluteness' and 'perfection'. While our
entry is focused on the strict mathematical sense of
'infinity', one cannot cleanly separate the various
meanings in the historical development of the subject, especially in
the first stages. In addition, treating infinity as a
'perfection' in theology from the outset does not mirror
the complexity of the historical development; for instance, we find
traces in the 13th century of thinkers who attributed finiteness to
God or in any case denied God's infinity even when not
explicitly stating the finiteness of God (see Cote 2002,
127-144).



The infinite has been of central concern to Western thought since the
very first pre-Socratic fragment. It concerned the philosopher
Anaximander (who flourished in the 6th century BCE), who identified
the principle and origin of existing things as *to apeiron*. In
Anaximander, the principle has both an ontological and an ethical
significance. The Pythagoreans (6th century BCE) saw the infinite
negatively and emphasized the lack of definiteness associated with it;
they also gave it spatial connotations. Indeed, in the 5th century BCE
the Pythagorean Archytas of Tarentum (see Huffman 2005, 540-550)
gave the following argument for the spatial infinitude of the cosmos
based on the contradiction that postulating a boundary to it would
seem to entail. If the cosmos is bounded, then one could extend
one's hand or a stick beyond its boundary to find either empty
space or matter. And this would be part of the world, which thus
cannot be bounded on pain of contradiction. So the world is
*unbounded*. Archytas identified this with the world being
*infinite*. Kant similarly identified the unbounded and the
infinite in his cosmological antinomy. In Section
 8
 we will see that these notions should be distinguished, but a
mathematically precise articulation of the distinction had to wait
until the development of new conceptions of space in the 19th
century.



The Eleatics (Parmenides and Melissus, 5th century BCE) held a monist
conception of reality, the One, and Melissus declared it to be
infinite. Such a monistic conception of reality sees change (or
becoming) as appearance, and Zeno's famous paradoxes of infinity
(see the entry on
 Zeno's paradoxes)
 emerge in this context. Suffice here to say that Zeno's
paradoxes (the Achilles, the arrow, and others) involved the
infinitely small and were aimed at buttressing Parmenides'
monism. Working across the 5th and the 4th century BCE, Democritus
defended an atomistic theory with an infinite void and infinitely many
atoms. The infinite by this time had shown some of its major aspects,
taken as substance by some and as plurality (of atoms, times,
geometrical points, etc.) by others.



If the urgency of problems related to the infinite reached Greek
consciousness with Zeno's paradoxes, the most influential
discussion was due to Aristotle. In order to put Aristotle's
discussion in perspective, we need to list a number of ways in which
mathematical infinity had emerged not only in philosophy, as we have
described, but also in mathematics. We have already seen with Archytas
the notion of spatial infinitude of the cosmos. But in number theory,
the natural numbers were considered infinite, at least in the sense
that given any natural number a greater one could be found. In
geometry, we find both the infinite by addition (any segment can be
extended) and by division (any segment can be halved). Thus,
mathematics presented processes of iteration without limit. The most
sophisticated technique for dealing with iterated processes in the
measurements of plane and solid figures was developed by Eudoxus (4th
century BCE), and we discuss it in Section
 2.1.



By the time Aristotle (4th century BCE) developed his discussion of
the infinite, this concept had thus made its presence felt in
philosophy, mathematics, and natural philosophy (including cosmology,
astronomy, and physics). It would be hard to exaggerate the role
played by Aristotle in the history of infinity. He articulated some
essential conceptual distinctions that were to influence all
subsequent discussions. He was a finitist in the sense that in his
universe, everything is finite. The cosmos is finite, bodies are
finite, geometrical segments are finite, each number is finite, etc.
However, there are processes that can be iterated indefinitely, giving
rise to what he called 'potential infinity'. He claimed in
fact that "in a sense [the infinite] is and in a sense it is
not." (*Phys.* 3.6, 206a13-14).



Any arbitrary segment can be extended in length (subject to
cosmological restrictions mentioned below) or halved without limit,
but at each stage we remain within the finite. Time is also
potentially infinite in both directions and can be divided without
limit.



This conception stands in opposition to that of 'actual
infinity', which would result if some infinite processes could
be completed, carried out 'all at once', as it were. If
actual infinity were real, then one could have infinitely long bodies,
infinitely long or infinitely small segments, the totality of natural
numbers, an infinite number, infinitely many instants of time, etc.
Aristotle rejected the notion of the infinite as a primordial
substance, as we have encountered in Anaximander, and most of his
discussion of the infinite takes place within a physical context,
namely one relating to spatio-temporal features of reality. As a
consequence, Aristotle's discussion of the infinite fell
squarely in what we have characterized as the
'mathematical' notion of infinity, where infinity applies
first of all to magnitudes (continuous or discrete) and what is
quantifiable (time, extension, numbers etc.). His *Physics*
discusses the infinitely large, excluded because the world is finite;
and the infinitely small, excluded because the division of matter can
only be potentially infinite and thus finite at each stage, never
reaching an infinitesimal quantity--one that is less than any
finite quantity, while being something. The exclusion of the
infinitely large also has as a consequence that Aristotle cannot allow
a potential infinity by addition in an unqualified manner (for
otherwise any finite extension could be added to itself sufficiently
many times to become larger than the size of the world). Infinity by
addition, then, is to be conceptualized as a sort of inverse operation
to infinity by division which gives us the primary evidence for the
existence of the potential infinite. This is the implicit force of the
contrastive "but" in the following quote. Aristotle writes
(our emphasis):



> 
> 'To be', then, may mean 'to be
> **potentially'** or 'to be
> **actually'**; and the infinite is either in
> **addition** or in **division**. It has been
> stated that magnitude is not in actual operation infinite; but it is
> infinite in division - it is not hard to refute indivisible
> lines - so that it remains for the infinite to be potentially.
> (*Physics* 3.6, 206a14-24)
> 



The Aristotelian distinction between potential and actual infinity has
had a major influence up to contemporary times. (For further
discussion of Aristotle on infinity see Hintikka (1966), Lear (1980),
Kouremenos (1995), Coope (2012), Nawar (2015), Cooper (2016), Ugaglia
(2018), and Hussey's commentary to Aristotle (1983).)



Aristotle's conception had, in addition to issues related to the
constitution of the physical continuum, important consequences in
cosmology. While he considered the cosmos to be finite, he thought
that the movement of the celestial spheres had no beginning and no end
(thus time for him, as we have noted, is potentially infinite in both
directions). The issue of the "eternity of the world" was
to exercise some of the best theological and philosophical minds after
Aristotle, especially in connection to theological issues. For
instance, Johannes Philoponus (6th century CE; see Philoponus 2004)
argued in favor of a beginning of the world by claiming that the
contrary thesis would lead to a paradox of infinity (we discuss this
in Section
 2.4).



Philoponus presented another paradox of infinity concerning infinite
time that we will discuss in the version formulated by
al-Ghazali (11th century CE)--see the



Supplement on al-Ghazali's objection




Of even more pressing significance was the abandonment of
Aristotle's view on the finiteness of the cosmos and the
Renaissance move from the finite to the infinite universe described in
the classic text by Koyre (1957; see also Jammer 1993). While
Copernicus (1473-1543) put the sun at the center of the
universe, he still worked with a finite model of the universe.
Foreshadowed by Epicurus (341-270), Hasdai Crescas
(1340-1412), and Nicolaus Cusanus (1401-1464), Giordano
Bruno (1548-1600) defended the idea of infinitely many worlds,
each of infinite size, existing simultaneously. Bruno is a good
example of how mathematical and theological notions of infinity were
used simultaneously in the history of the concept. For instance, in
*On the Infinite, the Universe, and Worlds* (1584) he argued
from God's infinite power to the infinitude of the universe.



By contrast, Kepler and Galileo did not think that the issue of
whether the world was infinite in size could be settled either way.
Kepler thought that the notion of an infinite universe was a
metaphysical one and not founded on empirical evidence. Galileo
claimed, in a famous letter to Francesco Ingoli written in 1624, that
mankind would never be able to know whether the universe is finite or
infinite. The progressive geometrization of space (see De Risi 2015)
led to Newton's gravitational theory in which the universe is
infinitely extended spatially and temporally. Physical space became
identified with the space of Euclid's geometry and in this way
physical space was geometrized.



Theological elements were still present when Newton identified space
with the "sensorium Dei" ("God's
sensorium"). For the next two centuries cosmology was developed
according to Newtonian theory: an infinite Euclidean space, flat and
absolute, which provides the receptacle for all physical objects whose
relations are structured by universal gravitation.



With Riemann in the mid-19th Century, and then with relativistic
cosmology, one went back to a finite universe, but cosmologists are
now fully aware that the issue of the finitude of the world is very
much an open question that depends crucially on the curvature and the
topology of space (see Section
 8.2).



Our discussion above indicates a few essential aspects of the concept
of infinity that will be useful in the later discussion. There are
obviously many areas of contact and/or intersection between the more
mathematical notion of infinity and the qualitative notion of
infinity. Qualitative notions of infinity cannot be easily
characterized directly but in general they appeal to features that do
not seem to have a clear quantitative aspect. For instance, God might
be defined as infinite because it has none of the limitations of
finite creatures; this property was accounted for in some Scholastic
philosophy by claiming that God, unlike finite creatures, is that
unique entity in which essence and existence coincide. Often coupled
with this was the claim that God's infinity is incomprehensible,
and this might be a good indicator that we cannot achieve a positive
account of qualitative infinity. At the same time, claims concerning
infinite divine power or goodness offer a possible connection to
quantitative conceptions, and this explains why the boundary line
between quantitative and qualitative conceptions is not so sharp.



Indeed, according to some authors the qualitative and mathematical
conceptions are inextricably tied. Consider for instance
Pascal's use of infinite distance both in projective geometry
and in his *Pensees* where he muses on the infinite
distance (and disproportion) between finite human beings and the
infinite God (see Cortese 2015). The following passage is
representative of the powerful and suggestive role that appeal to
finiteness and infinity plays in Pascal's apologetics:



> 
> For in the end what is humanity [l'homme] in nature? A
> nothingness compared to the infinite, everything [un tout] compared to
> a nothingness, a mid-point between nothing and everything, infinitely
> far from understanding the extremes; the end of things and their
> beginning [principe] are insuperably hidden for him in an impenetrable
> secret. \(\langle\)What can he therefore imagine? He is\(\rangle\)
> equally incapable of seeing the nothingness from where he came, and
> the infinite in which he is covered [englouti]. [...] (Pascal
> 2008: 70; we have added the French original where the translation
> seems less than faithful).
> 



Moreover, Pascal's *pari* (wager) is also intimately tied
to the notion of infinity in the form of an infinite reward. (See
Section
 7.3
 on Pascal's wager) These topics are of great importance for
philosophy of religion, decision theory, and philosophical
anthropology.



However, this entry does not concern those conceptions of infinity
that are connected to infinite divine power, infinite modes, and in
general about those conceptions of infinity that are not of a
mathematical kind. We do not intend to downplay the importance of
those aspects of the history of infinity to which giants such as
Plotinus, Cusanus, Descartes, Pascal, Spinoza, Fichte, Hegel, and
Kierkegaard contributed, among others. Leibniz and Kant also belong to
that list, but we will say more about them later on. But our entry
would lose focus if we were to try to pursue all these developments
even at a superficial level, and the treatment of qualitative infinity
is worthy of an article in its own right. Thus, we content ourselves
with a list of bibliographical references through which the reader can
reconstruct the contributions to the topic.



For overviews of the history of infinity which include both
mathematical and metaphysical aspects, see Moore (1990/2019) and
Zellini (2005). For further discussion of Aristotle's views on
infinity see the entries on:
 Aristotle;
 Aristotle and mathematics; and
 Aristotle and metaphysics.
 For ancient and medieval conceptions of infinity see Sweeney (1972),
Sweeney (1992), Kretzmann (1982), Cote (2002), Biard and
Celeyrette (2005), Duhem (1987), Dewender (2002), Davenport (1999),
Murdoch (1982), Uckelman (2015); for the early modern period see
Nachtomy and Winegar (2018); for infinity in Kant and the idealist
period see Kreis (2015); Monnoyeur (1992) spans all periods.



For more on infinity in philosophy of religion, see the following
references.


1. on divine infinity: Koetsier and Bergmans (2005), Gocke and
Tapp (2018), the papers in the final section of Heller and Woodin
(2011), and various entries including
 God and other ultimates,
 ontological arguments,
 Nicolaus Cusanus,
 Robert Grosseteste,
 John Duns Scotus,
 and
 Ibn Arabi;
2. on infinity in God's creation, apart from our subsequent
discussion of whether space and time are infinite: the entries
 cosmology and theology,
 cosmological argument,
 fine-tuning,
 infinite regress arguments,
 principle of sufficient reason,
 and
 being and becoming in modern physics;
 and
3. on 'heavenly infinity', apart from our subsequent
discussion of Pascal's Wager: the entries on
 Pascal's wager,
 the meaning of life, and
 religion and morality.



It is worth noting that Cantor's development of set theory was
influenced by theological considerations: see, for example, Dauben
(1990) and Tapp (2005).



As we have said, we are mostly excluding the topic of infinity in
science and the social sciences from our purview, although see the



Supplement on Infinite Idealizations.



To the extent that we discuss infinity in science (notably in Section
 8),
 our focus is primarily on the mathematical machinery involved, which
has a venerable history. This brings us to the topic of the next
section.


## 2. Infinity in mathematics: a brief historical overview



In this section we will begin by showing how Greek mathematics
studiously avoided the use of infinity in the presentation of its
results by making use of *the method of exhaustion* (3.1). Then
we will look at the widespread use of infinitary objects and
procedures in 17th-century mathematics (theory of indivisibles and
points at infinity in geometry (3.2), infinitesimals in the calculus
(3.3)) and Galileo's problem of extending counting to infinite
collections (3.4). By the early 18th-century mathematics had undergone
its first "infinitistic revolution" (the second is
associated with the name of Cantor, see section
 3).
 Infinity had become a pressing foundational problem, and this will
lead us to section
 3.


### 2.1 The method of exhaustion



We have already mentioned that the potential infinite occurs in Greek
mathematics from the outset, most obviously in the natural number
series and in the geometrical operations of addition and division of
segments and other geometrical magnitudes. The Greek mathematicians,
starting with Eudoxus, developed a technique for measuring plane and
solid figures that avoided recourse to the infinite even where an
infinite "limit" process would seem to be forced by the
situation. This technique, known today as *the method of
exhaustion* (the expression was coined in the 17th century by
Gregory of Saint Vincent), is found in Euclid's
*Elements*, book XII, and then in some of the most spectacular
results by Archimedes (3rd century BCE). The idea is to replace an
infinite approximation by a double reductio ad absurdum. That means
that one shows the equality in area or volume of two figures, say a
circle \(C\) and an associated triangle \(T\), by noting that \(C \lt
T, C \gt T\) or \(C = T\) and then showing that the assumptions \(C
\lt T\) and \(C \gt T\) both lead to a contradiction. (Here
'\(C\)' and '\(T\)' refer with systematic
ambiguity to the figures and their areas/volumes.) Further discussion
can be found in the



Supplement on Quadratures of the Circle by Exhaustion and by Indivisibles.



Greek mathematics generally avoids any recourse to the actual
infinite, and scholars have spoken of a "horror of
infinity" typical of Greek mathematics. This is in general
correct with respect to the way mathematical results are presented in
their final and public presentation. However, one should keep in mind
that no such "horror of infinity" is to be found when one
looks at the heuristic strategies pursued by Greek mathematicians. In
the case of Archimedes, this was made evident by the fortunate
rediscovery of his method (found in 1906; see Netz and Noel 2007)
where we see him using infinitary and mechanical considerations as
tools he exploited for the discovery of geometrical theorems (see
Knorr 1982, 1986 and Jullien 2015). For instance, in his description
of the method for finding the proportion between the area of a
parabolic segment and that of a related triangle, Archimedes thinks of
geometric figures (the parabolic segment and the related triangle, in
this case) as composed of infinitely many one-dimensional segments and
then exploits the law of the lever to gain the determination of the
relation between the areas in question. In a portion of the text of
the method that has only recently become available (a section of
proposition XIV, see Netz and Noel 2007), Archimedes explicitly
operates with infinite collections.


### 2.2 The theory of indivisibles and points at infinity



Early modern mathematicians were impressed by the Euclidean and
Archimedean rigor, but there was widespread suspicion (confirmed in
1906) that Archimedes must have had a less rigorous heuristic method
that he used to discover his surprising results.



In the 17th century, infinitary considerations in geometry opened the
way to new geometrical techniques in *quadratures* and
*cubatures*--i.e. the determination of areas of plane
figures and of volumes of solid figures, respectively. We owe to
Cavalieri and Torricelli a geometrical theory of indivisibles that was
later put in an arithmetico-algebraic setting by Wallis (1656).
Cavalieri's original idea (1635) was that the relation between
the areas of two plane figures could be obtained by a systematic
comparison of what he called the *indivisibles* of the figures.
An indivisible of a figure is a geometrical entity of lower dimension
than the figure itself. An indivisible of a line is a point; an
indivisible of a plane figure is a line segment; an indivisible of a
solid is a plane figure. Consider a square with top side AB and bottom
side CD. An indivisible of the square is any arbitrary segment with
the same length as AB that can be obtained by letting AB move parallel
to itself until it reaches CD. See the



Supplement on Quadratures of the Circle by Exhaustion and by Indivisibles




for an explanation of how to give the quadrature of the circle with
the indivisibilist method, and how this courts infinite
collections.



Cavalieri's applications of the theory of indivisibles were
limited to finite figures and thus did not go beyond the geometrical
boundaries typical of Greek mathematics. However, Torricelli broke new
ground with the determination of the volume of an infinitely long
(*infinite longum*) solid (Torricelli 1644). Up to then, all
the results concerning finite figures obtained through indivisibles
could easily be proved by finitary Archimedean techniques and by
avoiding any mention of infinity--just as in the case of the
quadrature of the circle presented in the



Supplement on Quadratures of the Circle by Exhaustion and by Indivisibles.



However, infinity figured explicitly in Torricelli's result that
an infinitely long solid (FEOBMDC in the diagram) had a finite volume
(the volume of the cylinder ACIH in the diagram).



![Illustration of the infinitely long solid studied by Torricelli. The solid is obtained by rotating a branch of a hyperbola around the y axis and extends to infinity. The illustration also displays a finite cylinder that equals in volume that of the infinitely long solid.](torricelli.png)


This was the first infinitary result in Western mathematics, for the
infinite was not eliminable using some alternative finitary technique
but rather showed up as a feature of the very object that had to be
measured. Torricelli's infinitary result put enormous pressure
on empiricist conceptions of infinity. The heuristic fruitfulness of
the indivisibilist method was also accompanied by paradoxes that
threatened its foundations. Among them was Tacquet's proof using
indivisibles that all triangles have the same area. The
indivisibilists were able to deal with such paradoxes in various ways,
but the foundations of the system remained shaky (for a detailed
discussion of the foundations of the theory of indivisibles and the
mathematical and philosophical issues connected to Torricelli's
result see Mancosu (1996) and Jullien (2015)).



Another area in which the infinite made its appearance in 17th century
geometry is in the work of Desargues (see Sakarovitch and Dhombres
1994 and Desargues 1636). Whereas in Euclidean geometry parallel lines
do *not* meet, Desargues entertained the idea of having
parallel lines meet at a point at infinity. This was a very fruitful
idea that led to the development of projective geometry.


### 2.3 The calculus



The most fruitful development in the use of infinity in 17th century
mathematics was that of the calculus.



From a geometrical point of view, the calculus provides techniques for
drawing tangents at an arbitrary point of a curve and for measuring
the area under a portion of a curve. The differential calculus treats
the first problem and the integral calculus the second. The
fundamental theorem of the calculus states that these problems are
inverses of each other. The calculus was developed independently by
Newton and Leibniz, but its spread owed much to a significant number
of mathematicians throughout Europe. The first textbook of the
differential calculus was published in 1696 by the Marquis de
l'Hopital (1696; see Bradley et al. 2015 for a
translation, which we follow below, with commentary). It is worthwhile
to consider its axiomatized structure, for it will help us see
immediately the infinitary foundations on which the new discipline
presented itself to the international community. We first have two
definitions:



**Definition I.** Those quantities are called variable
which increase or decrease continually, as opposed to constant
quantities that remain the same while others change.



**Definition II.** The infinitely small portion by which
a variable quantity continually increases or decreases is called the
Differential.



The two postulates are as follows.



**Postulate I.** We suppose that two quantities that
differ by an infinitely small quantity may be used interchangeably, or
(what amounts to the same thing) that a quantity which is increased or
decreased by another quantity that is infinitely smaller than it is,
may be considered as remaining the same.



**Postulate II.** We suppose that a curved line may be
considered as an assemblage of infinitely many straight lines, each
one being infinitely small, or (what amounts to the same thing) as a
polygon with an infinite number of sides, each being infinitely small,
which determine the curvature of the line by the angles formed amongst
themselves.



We see in the above the explicit infinitary characterization of some
of the basic entities appealed to in the new calculus. Both postulates
require something that the Greeks had studiously avoided, namely the
consideration of infinitely small quantities and the reduction of
curves to infinilateral polygons. While l'Hopital and a
number of French mathematicians were enthusiastic about going
"infinitary", Leibniz himself developed a fictionalist
account of the appeal to infinitely small quantities (foreshadowed
already in his early *De Quadratura* which did not see the
light of day until 1993; see Leibniz (1993)). Also note the use of
geometrical and kinematic (i.e. based on movement, as implied by the
notions of continual increase or decrease) concepts. Much of the 19th
century work on the calculus was devoted to removing geometrical and
kinematic notions from the foundations of the discipline.



The literature in this area is enormous and we refer to Goldenbaum and
Jesseph (2008) for a recent collection of essays on Leibnizian
infinitesimals. The debates on the foundations of the calculus led to
some lively contributions, such as Berkeley's *The
Analyst* (1734) and more mathematical work. But even after
infinitesimals were eliminated from the calculus through the combined
work of Cauchy, Bolzano, Dedekind, and Weierstrass in the 19th
century, they were widely employed in geometry. Moreover, contemporary
alternative theories of analysis (non-standard analysis, infinitesimal
analysis etc.) have led to rigorous theories that, taken with a grain
of salt, can be seen as vindicating some of the 17th century
intuitions. We will come back to these developments below.


### 2.4 Counting infinite collections



There is one final aspect of 17th century discussions of infinity that
is relevant for later considerations: the problem of extending the
concept of counting from the finite to the infinite. This problem is
related to the issue of whether there is only one infinity or whether
there might be different sizes of infinity. As we have mentioned,
Philoponus argued that the eternity of the world led to a
contradiction. In particular, he claimed, if the world has no
beginning in the past, then the number of individuals up to Socrates
would be infinite; but then by adding the number of individuals from
Socrates to now, one would obtain an infinity larger than the previous
one, and this, he concluded, is "one of the most impossible
things" (see Sorabji 1983). It was typical of Greek thought to
reject the idea that there can be different sizes of infinity.



The Islamic mathematician Ibn Qurrah (9th century CE) took a decidedly
infinitistic attitude and argued, against the Aristotelian
commentators, that there can be different sizes of infinity (see
Rashed 2009). He claimed, for instance, that the odd and the even
numbers have the same size, but that the multiples of three are 1/3 of
the total number of natural numbers. Contrary to what has been claimed
in the literature, his intuition was not that the even numbers and the
odd numbers have the same size because there is a one-to-one
correspondence between them. Rather we have a
"frequentist" intuition: every even number is followed by
an odd number; multiples of three appear every three numbers, etc. We
find a similar position in Grosseteste's treatise *De
Luce* (see Mancosu 2009, 2016 for an overview of the historical
developments and further references).



Galileo Galilei epitomized the paradoxical situation we run into when
trying to generalize counting from the finite to the infinite. In
*Two New Sciences* (1638; Galilei 1974), he presented a paradox
of infinity. On the one hand, there is an intuition that there are
*fewer* square numbers than natural numbers, since the first
collection is properly contained in the second (the former has some
but not all of the latter's members). On the other hand, there
is an intuition that there are *the same number* of squares and
natural numbers, since there is a one-to-one correspondence--a
*bijection--*between the natural numbers and their
squares. Galileo's own conclusion, following Oresme and Albert
of Saxony who had discussed similar issues in the 14th century, was to
claim that one cannot apply the relations of *equality*,
*greater than*, and *smaller than* to infinite
collections. Much subsequent theorizing about infinity can be regarded
as respecting one intuition at the expense of the other.



The intuition that if one set is a proper subset of another, the
former is smaller than the latter, traces back to Euclid--call
this the *part-whole* intuition. Bolzano (1851) was sympathetic
to it, and he tried to develop a theory of infinite sets that
preserved it. He was not successful, but he warned his readers not to
conflate one property of an infinite set--that it can be put in
one-to-one correspondence with a proper subset of itself--with a
criterion of 'size' (what he called the
"multiplicity" of a collection). Cantor (see Hallett
1986), by contrast, later used one-to-one correspondence as the
defining characteristic of *cardinal numbers*: the numbers that
answer 'how many?' questions, and that generalize counting
from the finite to the infinite in his set theory. He thus sided with
the intuition that if there is a bijection between two sets, they have
the same size--call this the *bijection* intuition. The
intuition is clearly correct for finite sets. For example, the set of
fingers on a normal human hand can be paired up with the set of toes
on a normal human foot, and vice versa: there is a bijection between
these two sets. And of course, the two sets have the same size (five).
A central question is whether the intuition is correct also for
infinite sets. We will discuss Cantor's theory and, by contrast,
some recent implementations of counting, known as theories of
*numerosities*, that preserve the part-whole intuition also for
infinite sets--see the



Supplement on Theories of Numerosities.



In conclusion, the "infinitistic revolution" in the 17th
and the early 18th century left an important legacy for philosophy and
mathematics. The theory of indivisibles introduced new magnitudes
characterized infinitarily (the collection of all the indivisibles of
a figure), and new infinitary geometrical objects extended the
classical geometrical universe. Moreover, the debates on the calculus
were focused on the nature of the infinitely small and the infinitely
large. Finally, the issues emerging from Galileo's paradox were
a prelude to the problem of extending counting from finite to infinite
collections.



These problems were gradually addressed in the 19th and 20th
centuries, and out of these discussions there emerged different
mathematical notions of infinity. We will work our way in stages to
these conceptions, via a discussion of some landmarks in the
contemporary mathematics of infinity.



Among the numerous general treatments of the use of infinity in
mathematics we recommend Levy (1987), Zellini (2005), Moore
(1990/2019), Vilenkin (1995), Barrow (2006). For more detailed
accounts of the history of the calculus, see Kline (1990), Boyer
(1959), Edwards (1979), and Grattan-Guinness (1980). The most recent
scholarship on the theory of indivisibles is to be found in Jullien
(2015). For recent collections on Leibnizian infinitesimals see
Goldenbaum and Jesseph (2008) and Goethe, Beeley and Rabouin (2015).
On concepts of mathematical infinity in the 19th century see
Konig (1990).


## 3. Mathematics: number systems, Cantor's paradise, and beyond



To deal with some of the issues concerning the infinite raised in
Section
 2,
 mathematicians have developed various different structures that
explicitly include infinities. These structures ascribe different
properties to infinities that are appropriate for different
applications. In some cases, there are multiple kinds of structure
that can be developed for an application. Explicitly countenancing
infinities has opened up an enormous range of choices and
possibilities, which has been a wellspring of development in modern
mathematics.



We now give a very quick tour of infinity in modern mathematics.
Section
 3.1
 reminds the reader of several familiar number systems: the natural
numbers, integers, rational numbers, and real numbers. Section
 3.2
 discusses the infinite operations of limits and sums that underlie
calculus, and introduces the "extended real numbers"
\(+\infty\) and \(-\infty\). The material in these two sections is
covered in most textbooks on real analysis, and even many calculus
textbooks, so some readers might already be familiar with it, while
others might benefit from having such a textbook on hand to expand on
some of the points.



Sections
 3.3
 and
 3.4
 are more mathematically advanced. Section
 3.3
 introduces Cantor's more mathematically sophisticated
"cardinals and ordinals", which are probably the
mathematical developments that have done the most to untangle many of
the conceptual confusions around infinity. This material is covered in
greater detail in most textbooks on set theory, and parts of it are
discussed in many logic textbooks as well. It can be read largely
independently of the other sections.



Section
 3.4
 discusses a more recent mathematical theory of infinitely large and
infinitesimal numbers that provides an alternate setting for calculus.
This theory of "non-standard analysis" has not become as
central a part of the mathematics curriculum as real analysis and set
theory. It may thus be unfamiliar to most readers, and it is harder to
find accessible introductions elsewhere. Although non-standard
analysis is not as central a part of the cultural understanding of
infinity in mathematics as cardinals are, we include it both because
it is a topic of growing interest in mathematics research, and because
it can help make mathematically rigorous sense of both many intuitive
thoughts about infinity and some of the early work on calculus in the
17th and 18th centuries.



Up to a point, various philosophical applications and puzzles
involving infinity can be understood without much understanding of the
mathematics of infinity. However, the mathematics helps us formulate
and tackle them rigorously. Mathematics-shy readers could skip parts
of our tour (particularly section
 3.4)
 and still benefit from the later sections, but we encourage them to
make the effort and read on. The mathematical understanding of
infinity is a great achievement in its own right.


### 3.1 Some number systems



The *natural numbers* form the most elementary number system.
(Some mathematicians count \(0\) as a natural number as well, but some
others do not.) \(1\) is a natural number. For any natural number
\(n\), \(n+1\)--the *successor* of \(n\)--is also a
natural number. The natural numbers--\(1, 2, 3, \dots\)--are
closed under addition: if \(n\_1\) and \(n\_2\) are natural numbers,
then so is \(n\_1+n\_2\). And they are closed under multiplication: if
\(n\_1\) and \(n\_2\) are natural numbers, then so is \(n\_1 \cdot n\_2\).
We use the natural numbers for counting 'how many' of
something there are, though they clearly fail when applied to infinite
sets, for instance the set of squares of natural numbers or the set of
natural numbers themselves. This is what 'infinities of
counting' will extend, in section
 3.3.



The *integers* consist of the natural numbers, their additive
inverses (a number and its additive inverse sum to \(0\)), and
\(0\):

 
\[\dots , -3, -2, -1, 0, 1, 2, 3, \dots.\]


They form the most elementary number system that is also closed under
subtraction: If \(j\_1\) and \(j\_2\) are integers, then so is
\(j\_1-j\_2\).



The *rational numbers* can be expressed in the form
\(j\_1/j\_2\), where \(j\_1\) and \(j\_2\) are integers, \(j\_2\neq 0\).
They form the most elementary number system that includes the integers
and that is closed under division, except by 0. If \(q\_1\) and \(q\_2\)
are rational numbers, \(q\_1/q\_2\) is also a rational number if
\(q\_2\neq 0\).



The rational numbers are *dense*: for any two rational numbers
\(q\_1\) and \(q\_2\) such that \(q\_1 < q\_2\), there is at least one
rational number \(q\_3\) such that \(q\_1 < q\_3 < q\_2\)--for
example, the arithmetic mean of \(q\_1\) and \(q\_2\), \((q\_1+ q\_2)/2\),
lies between them. Indeed, for any two rational numbers \(q\_1\) and
\(q\_2\) such that \(q\_1\) is strictly less than \(q\_2\), for any
natural number \(n\), there are more than \(n\) distinct rational
numbers that lie between \(q\_1\) and \(q\_2\). Where the integers
spread infinitely 'outward' in both directions, the
rationals also divide infinitely 'inward'.



However, the rational numbers still have "gaps". For
instance, if we consider the equation, \(y=x^3-2\), we can verify that
there are values of \(x\) where \(y\) is negative, and values of \(x\)
where \(y\) is positive. However, there is no rational number \(x\)
for which \(y\) is exactly equal to 0. To fill these gaps, we
construct the "real numbers".



The real numbers can be constructed out of the rational numbers by
defining each real number to be a *Dedekind cut* of the
rationals. A Dedekind cut of the rationals is a pair of sets \(L\) and
\(R\) such that:


1. every rational number belongs to exactly one of \(L\) and \(R\);
2. every member of \(L\) is less than each member of \(R\); and
3. \(L\) has no largest element.



We refer to \(L\) as the 'left set' of the cut and \(R\)
as the 'right set'.



For any rational number \(q\), there is a Dedekind cut corresponding
to it, where \(L\) consists of the numbers strictly less than \(q\),
while \(R\) consists of \(q\) and all larger numbers. However, there
are also other partitions, where \(R\) does not contain a smallest
element. For instance, we can let \(L\) include all the rational
numbers whose cube is less than 2, while \(R\) includes all the
rational numbers whose cube is greater than 2. Since there is no
rational number whose cube is exactly equal to 2, this pair of sets
forms a partition, representing the real number we think of as the
cube root of 2.



If \(x\) and \(y\) are two real numbers, represented by Dedekind cuts
with left sets \(x\_L\) and \(y\_L\), and right sets \(x\_R\) and
\(y\_R\), we can define operations of addition and multiplication of
real numbers in terms of operations on the members of these sets. The
left set of \(x+y\) is the set of all rational numbers that result
from adding a member of \(x\_L\) and a member of \(y\_L\), while the
right set is the set of all rational numbers that result from adding a
member of \(x\_R\) and a member of \(y\_R\). (It takes a little work to
check that every rational number is in fact in one of these two sets,
but the other conditions for being a Dedekind cut are straightforward
to check.) If \(x\) and \(y\) are both positive, then we can define
the right set of \(x\cdot y\) as the set of all rational numbers that
result from multiplying a member of \(x\_R\) by a member of \(y\_R\),
with the left set defined as the set of all other rational numbers.
(Some modifications of this definition are needed if either \(x\) or
\(y\) is negative.) Subtraction and division can then be defined as
the inverses of these operations, just as for the rationals.



The real numbers are closed under addition, subtraction,
multiplication, and division by all real numbers except 0. They have
the further feature that there are no "gaps": for any
bounded set of real numbers, there is a least upper bound, and for any
continuous function from real numbers to real numbers, if the function
is negative at one point and positive at another, there must be some
point at which it is exactly equal to 0. For further discussion, see
the entry on
 Dedekind's contributions to the foundations of mathematics.



There are several uses for which we want a number system with no gaps,
and thus we use the real numbers. If you try to measure the amount of
water in a large vessel by counting out a specific number of small
cups, there's no guarantee the number of cups will be an
integer. If you try to measure a long distance by counting out the
length of your foot, there's no guarantee the number of feet
will be an integer. We might know that something is more than 4 of the
units and less than 5. By moving to fractions of these units, we can
get more precise--4 cups and 5 to 6 ounces, or 4 feet and 5 to 6
inches--but there's still no guarantee that a specific
rational number will give the precise amount. But we can generate a
sequence of approximations that get closer and closer by using smaller
and smaller fractions of these units. So in measuring 'how
much' of something there is, or giving coordinates to describe
the location of a point in geometric space, we use the real numbers,
to guarantee, as we show in the next section, that there is some
precise value the sequence of approximations converges to.


### 3.2 Limits, infinite sums, and the extended real numbers; \(+\infty\) and \(-\infty\)



The mathematical property of 'lacking gaps' is referred to
as 'completeness' -- the formal statement is that an
ordered set is *complete* if every bounded, increasing sequence
of elements has a 'limit'. These limits are the first
infinite operation that we define on numbers.


#### 3.2.1 Limits of sequences



A *sequence* of numbers is an ordered list of numbers, which we
may symbolize:

 
\[a\_1, a\_2, \dots\]


or

 
\[\langle a\_n\rangle.\]


The members of a sequence are indexed by the natural numbers, \(n=1,
2, \dots\).



The formal definition of a limit says the sequence \(\langle
a\_n\rangle\) *converges to \(l\)*, or *has limit \(l\)*
if and only if the terms of the sequence eventually stay arbitrarily
close to \(l\). Formally:

 
\[\lim\_{n\to\infty} a\_n=l\]


iff



> 
> 
> for each real number \(\epsilon>0\), there exists a natural number
> \(N\), such that for every natural number \(n>N\),
> \(|a\_n-l|<\epsilon\).
> 
> 
> 



We will say a bit more about the symbol '\(\infty\)' that
appears in this definition later, but for now it just indicates the
behavior of the sequence beyond any particular finite point.



As we will demonstrate shortly, not every sequence has a limit, but we
can define an important class of sequences that do. A sequence is
*increasing* if each term in the sequence is at least as large
as the previous term. A sequence is *bounded* if there is some
real number that is larger than every term in the sequence. It turns
out that every bounded, increasing sequence has a limit. The
successive approximations to the measurement of some physical quantity
with a finer and finer measuring unit will amount to a bounded,
increasing sequence of numbers, and thus this definition of a limit
allows us to give a numerical representation of any physical
quantity.



To show that every bounded, increasing sequence has a limit, consider
the Dedekind cuts defining the individual real numbers in the
sequence. Let us define a new Dedekind cut by taking its left set to
contain every rational in the left set of at least one of these terms
of the sequence, and taking its right set to contain every rational
that is in *all* of the right sets of these terms. Because the
sequence is bounded, we know that the right set is non-empty, and the
rest of the properties of a Dedekind cut are not hard to check.



It is not hard to check that the real number constructed in this way
is the limit of the sequence. To see how this works in a specific
case, we can consider the sequence \(1/2, 3/4, 7/8, \dots, 1-1/2^n,
\dots\) This sequence is increasing, since each term is greater than
the one that came before, and it is bounded, since 2 is a number that
is strictly greater than every term in the sequence. The Dedekind cut
constructed as above will correspond to the number 1. To see this,
note that for any rational number q less than 1, we can let
\(\epsilon=1-q\). Then there is some \(N\) such that
\(1/2^N<\epsilon\). The \(N\)th term in the sequence will then be
greater than \(q\), so \(q\) will be in its left set, and thus in the
left set we constructed above. But 1 itself, and every rational number
greater than it, are all in the right set constructed above. This
reasoning also shows that the sequence converges to 1 according to the
definition of a limit. For any \(\epsilon\), and an \(N\) such that
\(1/2^N<\epsilon\), the \(N\)th term in the sequence is within
\(\epsilon\) of 1, and all later terms of the sequence are greater
than the \(N\)th term, but still less than 1, so they must also all be
within \(\epsilon\) of 1.



This fact about bounded, increasing sequences also underlies the use
of infinite decimal notation for real numbers. When we say that the
number \(\pi=3.1415926\dots\), we just mean that \(\pi\) is the limit
of the sequence \(3, 3.1, 3.14, 3.141, \dots\). One fact about this
notation that many people find surprising is that the decimal notation
\(0.99999\dots\) is the limit of the sequence \(.9, .99, .999,
\dots\), and thus is precisely 1. Some people feel the intuition that
\(0.9999\dots\) should somehow denote a number 'infinitely
close' to 1, but not equal to it. We will be able to make sense
of an idea like this in section
 3.4,
 but it turns out that decimal notation is not the way to do it. For a
useful demonstration of this point, see Vi Hart's video
 9.999... reasons that .999... = 1.



Many sequences that are not increasing have limits as well. For
instance, the sequence \(1, -1/2, 1/3, -1/4, \dots, (-1)^n/n, \dots\)
can be seen to converge to the value 0, even though it is not
increasing. However, if a sequence is not bounded, like the sequence
\(1, 2, 3, \dots\), it does not have a limit--if it had a limit,
there would have to be some values \(l\), \(\epsilon\), and \(N\),
such that all terms in the sequence beyond the first \(N\) are within
\(\epsilon\) of \(l\). But then any number that is larger than the
first \(N\) terms of the sequence, and also larger than
\(l+\epsilon\), would be a bound for the sequence. And some sequences
that are not increasing also fail to have a limit--for instance,
the sequence \(1, 0, 1, 0, 1, 0, \dots\) does not have a limit,
because there is no value such that all terms of the sequence are
eventually within \(1/3\) of that value.


#### 3.2.2 Infinite sums and products



With the definition of the limit of a sequence, we can now also often
define infinitary versions of the operations of addition and
multiplication. For a finite number of terms, we define the
'partial sum' \(\sum\_{i=1}^n a\_i=a\_1+\dots+a\_n\) and
'partial product' \(\prod\_{i=1}^{n} a\_i=a\_1\cdot\dots\cdot
a\_n\). For an *infinite* sequence of numbers \(a\_n\), their
infinite sum or product (when it is defined) is the limit of the
partial sums or products:

 
\[\sum\_{i=1}^\infty a\_i=\lim\_{n\to\infty}\sum\_{i=1}^n a\_i\]


and

 
\[\prod\_{i=1}^\infty a\_i=\lim\_{n\to\infty}\prod\_{i=1}^n a\_i.\]


Thus,
\(\sum\_{i=1}^\infty\frac{1}{2^i}=\lim\_{n\to\infty}\sum\_{i=1}^n\frac{1}{2^i}=\lim\_{n\to\infty}(1-\frac{1}{2^n})=1\).
We can show that if an infinite sequence of terms has a sum that
converges, then the terms themselves must converge to 0. That's
because the terms \(\sum\_{i=1}^n a\_i\) must converge, so that for any
\(\epsilon\), there must be \(N\), such that every one of the
\(\sum\_{i=1}^n a\_i\) is within \(\epsilon\) of the limit whenever
\(n>N\). Thus, every such \(a\_n\) must have absolute value less
than \(2\epsilon\), so that both successive partial sums can be within
\(\epsilon\) of this limit. (A similar condition holds also for
infinite products, but from now on we will focus only on infinite
sums.)



However, just having the terms \(a\_n\) converge to 0 is not sufficient
for the sum to converge. A deep and important fact about infinite sums
is that \(\sum\_{i=1}^\infty\frac{1}{i}\) fails to converge, because
the partial sums eventually exceed any finite bound. To see this, note
that the first term is greater than \(1/2\), the next two terms are
both greater than \(1/4\), the next four terms are all greater than
\(1/8\), and in general there are \(2^{n-1}\) terms that are each
greater than \(1/2^n\). So to get a partial sum greater than \(n\), it
is sufficient to add the first \(2^{2n}\) terms.



But if the terms \(a\_n\) converge to 0, and each one is smaller in
absolute value and has the opposite sign as the previous term, then
the infinite sum must converge. If the first term in the sequence is
positive, this is because the even numbered partial sums form a
bounded, increasing sequence, and the odd numbered partial sums form a
bounded, decreasing sequence, and the difference between successive
terms of these two sequences is a term of the original sequence and
thus converges to 0. Thus, the sum
\(\sum\_{i=1}^\infty\frac{(-1)^{i+1}}{i}\) converges (in particular, to
the natural logarithm of 2). But a somewhat surprising fact, known as
the Riemann rearrangement theorem, states that if the positive terms
of a sequence have no finite sum, and the negative terms of the
sequence have no finite sum, but the individual terms themselves
converge to 0, then for any real number \(x\), the terms of the
sequence can be put into some order so that \(x\) is the sum of the
sequence in that order! To prove this, just rearrange the terms by
beginning with enough positive terms to bring a partial sum above
\(x\), then enough negative terms to bring a partial sum below \(x\),
then enough partial sums to bring a partial sum above \(x\) again, and
so on. This process must be able to be carried out, because the
partial sum of the positive terms eventually exceeds any finite bound,
and similarly with the partial sum of the negative terms. Since the
individual terms of the sequence are all eventually within
\(\epsilon\) of 0, these partial sums must eventually never differ
from \(x\) by more than \(\epsilon\), and so the sum of this
arrangement of the terms must converge to \(x\).



Thus, infinite summation has some importantly different features from
finite summation. For any finite set of real numbers, the sum of those
numbers is well-defined, and doesn't depend on the order you add
them. But with an infinite sequence of real numbers, there may be no
number that is the sum of that sequence in that order. And even if
there is, it may be possible to rearrange the terms of the sequence so
that they sum to another value.



But there are some cases in which the sum can be known to behave
nicely. If all the terms in the sequence are positive, and there is
*some* order in which their sum converges, then their sum must
converge to this value in *any* order. This is because the
partial sums form an increasing sequence, and for any two orderings of
the terms, and any partial sum of one of those orderings, there must
be some partial sum of the other ordering that includes all of the
terms in that partial sum, and vice versa. Similarly, the value of the
sum doesn't depend on the order of summation if all the terms
are negative. And if the positive terms have a convergent sum, and the
negative terms of a series also do, then the sum of the series taken
in any order must be equal to the sum of these two sums. Such a
sequence is said to be *absolutely convergent*, as the sum of
the absolute values of the terms converges.


#### 3.2.3 Limits of functions and the extended real numbers \(\pm\infty\)



Infinite sequences and sums aren't the only ways that limits
appear in mathematics. Functions of a real value can also have limits.
The limit of a function of a real value \(x\) as \(x\) goes to
infinity is defined in a similar way to the limit of a sequence
indexed by natural numbers. To say

 
\[\lim\_{x\to+\infty} f(x)=l\]


is to say that for every \(\epsilon\), there is an \(N\), such that
whenever \(x>N\), \(f(x)\) is within \(\epsilon\) of \(l\). For
example, \(\lim\_{x\to+\infty}e^{-x}=0\), because \(e^{-x}\) can be
made as small as one likes by choosing large enough \(x\). (Note that
the inputs to a function can be positive or negative, so we need to
specify that \(x\) approaches *positive* infinity to
distinguish this from the limit at the other end of the axis.)



But it is also often useful to be able to define the limit of a
function at some particular finite real-valued input. For instance, we
might be interested in the function \(f(x)=\frac{x^2-9}{x-3}\) as
\(x\) approaches 3. (As we will see in section
 3.4,
 this sort of calculation is particularly important in defining the
concept of 'derivative' of a function, giving the slope of
a continuous curve at a point.) This particular function is defined
for all real numbers *other* than 3, and at any such input
\(x\) it takes the value \(x+3\). We would like to be able to say that
the limit of this function as \(x\) approaches 3 is 6. The way we make
this precise is to say that

 
\[\lim\_{x\to a} f(x)=l\]


iff



> 
> 
> for each real number \(\epsilon>0\), there exists a \(\delta\),
> such that for every \(x\) with \(0<|x-a|<\delta\),
> \(|f(x)-l|<\epsilon.\)
> 
> 
> 



That is, for any degree of approximation to the limit that we want,
there is some degree of approximation for the input that suffices to
guarantee that the function is that close. In the initial definition
of a limit as \(x\) goes to \(+\infty\), we required \(x\) to
'approximate' \(+\infty\) by being sufficiently large, but
now we require it to approximate \(a\) by having a difference from
\(a\) that is sufficiently small, just as the values of the sequence
or function approximate the limit \(l\).



We can do the reverse to put \(\infty\) on the right side of the limit
as well. That is, we say

 
\[\lim\_{x\to a} f(x)=+\infty\]


iff



> 
> 
> for every \(M\), there exists a \(\delta\), such that for every \(x\)
> with \(0<|x-a|<\delta\), \(f(x)>M\),
> 
> 
> 



and similarly

 
\[\lim\_{x\to+\infty} f(x)=+\infty\]


iff



> 
> 
> for every \(M\), there exists an \(N\), such that for every
> \(x>N\), \(f(x)>M\).
> 
> 
> 



Since \(\infty\) (or more precisely
'\(+\infty\)'--similar methods work for
'\(-\infty\)') can appear in each place where a real
number can appear in this limit notation, it is natural to see if we
can extend the definition of real numbers so that it can be
included.



And in fact, if we take the definition of a Dedekind cut, and relax
the requirement that the left and right sets be non-empty, we get two
new elements--the one with an empty right set is greater than
every rational number, and called '\(+\infty\)', while the
one with an empty left set is less than every rational number, and
called '\(-\infty\)'. Within these 'extended real
numbers', not only does every *bounded* increasing
sequence have a limit, but *every* increasing sequence has a
limit.



Just as the real numbers emerge naturally as the tools to measure
finite quantities, as the limits of rational approximations, the
extended real numbers emerge naturally as the tools to measure
potentially infinite quantities that can be approximated by finite
quantities. \(+\infty\) can be taken to be the area of an infinite
region, the length of an infinite line, the limit of \(1/x^2\) as
\(x\) goes to 0, and so on. Although we are used to thinking of
lengths and areas as positive numbers, it is sometimes useful to
consider them as negative as well, when we care about the direction
that they are pointing, and in this sort of situation \(-\infty\) is
useful as well. Just as the real number operations of addition,
multiplication, subtraction, and division correspond to certain
operations on the quantities they measure, these operations can often
be extended to these extended real numbers, as long as we are careful
about a few cases.



Adding or subtracting a finite area from an infinite area leaves it
infinite. Adding a shape with infinite area to another shape of the
same infinite area leaves the total area unchanged, and subtracting a
negative infinite one from a positive one or vice versa is similar.
But \((+\infty) + (-\infty)\) cannot be meaningfully evaluated; nor
can \((+\infty) - (+\infty)\). If you take one infinitely large
region, and remove an infinitely large region, you might be left with
nothing, or a positive region, but you might still be left with an
infinitely large region; or if the region you subtracted was larger
than the original region, you might be left with a negative
region--even an infinite negative region.



These restrictions also apply to the use of these extended real
numbers as the limits of sequences or functions. Whenever two
sequences or functions both have a finite limit, their sum or
difference will have a limit that is the sum or difference of their
limits. When one has a finite limit and the other is infinite, then
their sum or difference will be determined by the one that is
infinite. But when both are infinite, there are problems. We can see
that \(1/x^2\), \(2+1/x^2\), and \(1/x^4\) are all functions that go
to \(+\infty\) as \(x\) goes to 0. If we add or subtract any of these
functions from a function with a finite limit, the resulting function
still has limit of \(+\infty\). If we add them to each other in any
combination, the result still has limit \(+\infty\). But if we
consider their differences, we see that \(1/x^2 - 1/x^2\) has 0 as its
limit, \(1/x^2 - 1/x^4\) has \(-\infty\) as its limit, and \(1/x^2
-(2+1/x^2)\) has \(-2\) as its limit. So "\(\infty -
\infty\)" is said to be an "indeterminate form" that
can't be evaluated.



Multiplying or dividing an infinite number by a finite positive number
leaves it unchanged, and multiplying or dividing by a finite negative
number reverses its sign. Similarly for multiplying the infinite
numbers by themselves or each other. But an infinite number divided by
an infinite number, or an infinite number multiplied by 0, are also
indeterminate forms. As \(x\) approaches 0, the function
\(\frac{1/x^2}{1/x^2}\) has limit 1, while the function
\(\frac{1/x^2}{1/x^4}\) has limit \(+\infty\). If we take the function
\(1/x^2\) whose limit is \(+\infty\) and multiply it by the function
\(x\) whose limit is 0, we get the function \(1/x\), that has no limit
as \(x\) approaches 0 (since it takes on both large positive and large
negative values in any small interval around 0--this is why we
have used \(1/x^2\) and \(1/x^4\) as the paradigms of functions with
limit \(+\infty\), rather than \(1/x\) or \(1/x^3\)). For similar
reasons, these extended real numbers don't provide a way to
divide by 0. So although the extended real numbers have some nice
properties, and can be used for measurement in various cases,
arithmetic involving them is not as nice as the standard real
numbers.


#### 3.2.4 Related infinities



The Dedekind cut construction was done as a way to make sense of
limits of the rational numbers. This first created the real numbers,
which can be thought of as the limits of bounded infinite sequences of
rational numbers. We then considered *all* limits that made
sense, including towards the ends of the real line, yielding the
extended reals, including the standard reals as well as \(+\infty\)
and \(-\infty\).



Versions of this process can be carried out with other mathematical
entities as well. Projective geometry adds additional points 'at
infinity' to the Euclidean plane, one for each family of
parallel lines, to help explain features of visual geometry, like the
way that parallel railroad tracks appear to meet at the horizon,
infinitely far away. Riemannian geometry extends the complex numbers
by adding just a *single* number \(\infty\) that one can
approach simultaneously 'in all directions' in the complex
plane. These alternate geometries provide foundations for material
discussed in section
 2.2,
 and also section
 8.2.
 Several of these are discussed in the entry on
 nineteenth century geometry,
 and others are discussed in topology textbooks under the topic of
'compactification'.



Because these infinities are inherently considered as limits of finite
approximations, there is no way for one infinite element to lie
"beyond" another--at most it can lie in a
"different direction", the way that the points of
convergence of different families of parallel lines do, or the way
\(+\infty\) and \(-\infty\) do in the extended reals.



But as we will shortly see, there are other notions of infinity for
which one infinity can lie "beyond" another. In section
 3.3,
 we will discuss the ideas of infinity that we get from generalizing
the use of the natural numbers for counting, rather than generalizing
the use of the real numbers for measuring. And in section
 3.4,
 we will discuss yet another mathematical theory of infinity, which
arises from an alternate formulation of calculus, where the
\(\epsilon\)'s and \(\delta\)'s are treated as actually
being infinitely small, rather than just being arbitrary finite
measures of smallness.


### 3.3 Infinities of counting


#### 3.3.1 Preliminaries



As mentioned above, this section is more mathematically dense than the
previous two. However, we need this level of mathematical rigor to
develop Cantor's theory of ordinals and cardinals, which are
widely regarded as the most significant mathematical advance in our
understanding of the infinite.



An insomniac, counting imaginary sheep to try to get to sleep, will
never run out of natural numbers to do the job: 1, 2, 3, ....
There is no bound on the set of natural numbers. This is our first
infinite set. It is perhaps a natural thought that there is just one
infinity for counting infinite sets, which we might symbolise
'\(\infty\)'. The thought may seem even more natural when
we *define* an infinite set as one that has the same size (in a
sense to be made precise shortly) as a proper subset of it. In fact,
the thought could hardly be more mistaken: as we will soon see,
according to mathematical orthodoxy--namely contemporary set
theory and the attendant notion of cardinality of a set--there
are *infinitely many infinities*. This prompts a series of
questions: Is there a smallest one? Yes, as we will see. Is there a
largest one? No, as we will see. What can be said about the spacing
between the infinities and how far infinities extend? Well, we will
see. We can also ask parallel questions about the infinitely
small.



Recall Galileo's paradox of infinity, based on the collision of
the part-whole intuition and the bijection intuition, and his
conclusion that one cannot apply the relations of 'less
than', 'equality', and 'greater than' to
infinite collections. Modern mathematical orthodoxy, embodied in
contemporary set theory, rejects Galileo's conclusion. That
orthodoxy is grounded in the bijection intuition, following Cantor
rather than Euclid and Bolzano. When there is a bijection between two
sets, we say that they have the same *cardinality*. The notion
of cardinality does not respect the part-whole intuition. For example,
the squares of natural numbers are a proper subset of the natural
numbers, but they have the same cardinality since they can be put in
one-to-one correspondence.



Foundational programs such as neo-logicism also start from a notion of
"equinumerosity" based on the Cantorian bijection
intuition. It was not until the early 2000s that a group of
mathematicians working on non-standard analysis (Benci, di Nasso, and
Forti) developed a theory of "numerosities" that agrees on
finite sets with Cantorian cardinality but that also upholds the
part-whole intuition for infinite sets, and thus diverges from
Cantorian cardinalities. (See Benci and Di Nasso 2003, 2019; Benci, Di
Nasso and Forti 2006, 2007). One can consider numerosities as a
refinement of Cantorian cardinalities. Every two sets that have the
same numerosity have the same cardinality, but the converse does not
hold. For instance, in this approach the set of squares has numerosity
strictly less than the set of natural numbers. See the



Supplement on Theories of Numerosities.


#### 3.3.2 Set theory: \(\omega\) and \(\aleph\)



The sort of infinities that are most familiar to philosophers are
infinities used for counting. In the early 1820s, Bolzano arrived at
the idea that an infinite set is one for which there is a bijection
between the set and a proper subset of it. (Recall Galileo's
paradox.) Dedekind (1884) gave this as a *definition* of being
infinite. It is easy to prove that a Dedekind-infinite set must
contain a set that is just as large as the natural numbers. See
the



Supplement on Proofs of Theorems.



Dedekind's definition is only one among several alternative
definitions of infinite set (and correspondingly of finite set) that
have been proposed by him and after him. If one assumes the Axiom of
Choice, these alternative definitions turn out to be equivalent. (This
axiom says that for every set \(A\) of pairwise-disjoint non-empty
sets, there exists a function that selects exactly one element from
each set in \(A\).) But without the Axiom of Choice, the definitions
can be shown not to be equivalent and the foundational situation is
rather delicate but well understood (see Moore 1982).



The modern theory of infinities of counting derives from Cantor
(1932). He observed that a natural way to set up a bijection between
finite sets is to order the elements of each set, and pair the first
element of one set with the first of the other, the second of one set
with the second of the other, and so on. This sometimes works with
infinite sets--for example, it gives the one-to-one
correspondence between the natural numbers and the squares (considered
under their standard ordering). When there is a one-to-one
correspondence between two sets, such that every pair of elements of
one set bears the same ordering as the corresponding pair of elements
of the other set, the two ordered sets are said to have the same
*order type*.



But for some infinite sets (notably the set of *all* integers,
including the negative numbers, and also the set of rational numbers)
there is no first element under their standard ordering. In this case,
it is possible to re-order the elements of the set so that every
non-empty subset has a first element, so that this process works. Such
an ordering is called a *well-ordering*. (That *every*
set can be well-ordered is, as Zermelo famously proved in 1904,
equivalent to the Axiom of Choice.)



We can reorder the integers, alternating between positive and
negative: 0, -1, 1, -2, 2, -3, 3, .... This ordering has the same
order type as the natural numbers, and thus enables a one-to-one
correspondence between the natural numbers and the integers. One of
Cantor's most striking early observations is that the same is
possible with the positive rational numbers. Every positive rational
number can be written uniquely in lowest terms as some fraction
\(p/q\), where \(p\) and \(q\) are positive integers with no common
factor. We can then order these fractions by first comparing the sum
\(p+q\) of numerator and denominator, and then if the sum is equal for
two fractions, put the one with lower numerator \(p\) first. This
ordering begins 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, 4, 1/5, 5, 1/6, 2/5,
3/4, 4/3, 5/2, 6, .... (Note that fractions like 2/2, 2/4, 3/3,
4/2, are missing from this list, because they are not written in
lowest terms.) Every positive rational number must appear in this list
(because it can be written in lowest terms with some particular finite
sum of numerator and denominator), and only has finitely many
predecessors (because there are at most \(n+1\) fractions whose sum of
numerator and denominator is \(n\)).



![An infinite two-dimensional grid including all the rational numbers, sorted by numerator in the rows and denominator in the columns, with arrows indicating a looping path that reaches every rational number in finitely many steps.](rational-ordering.png)

The ordering: 1, 1/2, 2, 1/3, 3, 1/4, 2/3, 3/2, .... Every
rational number is eventually included.




However, Cantor also observed that different well-orderings of the
same infinite sets will produce a different order type. For instance,
we can define an ordering on the integers where every non-negative
integer comes before every negative integer, with any two integers of
the same sign being ordered by their absolute value. To approximately
represent this ordering we could write it as \(0, 1, 2, 3, \dots, -1,
-2, -3, \dots\). In this ordering, every non-empty subset still has a
first element (the non-negative element of lowest absolute value if it
contains any non-negative elements, and the negative element of lowest
absolute value if it only contains negative elements). If we pair up
the first element of this ordering with the first of the standard
ordering on the natural numbers, and the second with the second, and
the third with the third, and so on, then the negative integers would
not be paired with any natural number. But we can put the natural
numbers into the same order type by declaring the odd numbers to be
before the even numbers, and ordering them by size within these two
sets: \(1, 3, 5, \dots, 0, 2, 4, \dots\). A single infinite set can be
given orderings of many different order types, and also different
orderings of the same order type (for instance if we put the even
numbers first and the odd numbers second).



Cantor noted that for any two well-ordered sets, the initial positions
in one ordering (the first, the second, the third, etc.) correspond to
the initial positions in the other, the way that they do for finite
sets. In fact, he showed that *all* of the positions of one
well-ordering must correspond to initial positions in the other. (If
this weren't true, then the set of positions in one that
don't correspond to positions in the other would be non-empty
for each set, and the first elements of these sets would correspond,
which would contradict the claim that these positions don't
correspond.) Thus, there is a single list of all the possible
positions in well-ordered sets, beginning with the first, second,
third, and so on, and these positions are called the *ordinal
numbers*. A well-ordered set can be said to have its own ordinal
number, which is the first ordinal number that does not correspond to
a position in that set.



A *cardinal number* (like "one", "two",
"three")--also called a
*cardinality*--represents how many elements a set has. Two
sets have the same cardinal number if it is possible to come up with
any correspondence at all between the members of one and the members
of the other, even if this correspondence fails to respect the
ordering or any other structure of the sets. Two *finite* sets
have the same cardinal number if and only if they have the same
ordinal number. For infinite sets, if they are well-ordered and have
the same ordinal number, then they have the same cardinal number
(because two sets well-ordered with the same order type have a unique
correspondence between elements in corresponding positions of the
ordering). But they may have the same cardinal number without having
the same ordinal number: we have seen that sets of one cardinality can
be represented with many different order types.



Cantor used lowercase Greek letters to represent infinite ordinal
numbers, with \(\omega\) representing the order type of the standard
ordering of the natural numbers. Addition of ordinal numbers
corresponds to the order type that results from taking an ordering of
the first type, followed by an ordering of the second type. Thus,
\(\omega+\omega\) represents the order type of the integers with the
non-negative numbers first and the negative numbers afterwards, while
\(\omega+1\) represents the order type of the natural numbers with
just one element put at the end. Note that \(1+\omega\) is the order
type of a single element, followed by a copy of the natural numbers,
which is in fact the same as the order type of the natural numbers!
Thus, \(1+\omega=\omega\), which is not equal to \(\omega+1\). So
ordinal addition is not commutative.



Von Neumann (1923) defined a canonical representation of ordinal
numbers, using the fact that the ordinals are themselves well-ordered.
Each ordinal is represented by the set consisting of all smaller
ordinals. Thus 0 is represented by the empty set \(\emptyset\), 1 by
the set containing the empty set \(\{\emptyset\}\), 2 by the set
containing both of those \(\{\emptyset,\{\emptyset\}\}\), and so on.
\(\omega\) is then the set containing all of these finite ordinals
\(\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\dots\}\),
\(\omega+1\) is the set containing it as well
\(\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\dots,\omega\}\),
and so on.



Multiplication of ordinal numbers corresponds to replacing each
element of the second ordering by an entire ordering of the first
type. \(\omega\cdot\omega\) represents the order type of an
\(\omega\)-sequence of \(\omega\)-sequences, which we can have by
taking the positive rational numbers and sorting them first by
denominator, and then by numerator: 1, 2, 3, ... 1/2, 3/2, 5/2,
..., 1/3, 2/3, 4/3, 5/3, ..., 1/4, 3/4, 5/4, .... Note
that \(\omega\cdot 2\) is 2 copies of \(\omega\), while
\(2\cdot\omega\) is \(\omega\) copies of 2. \(\omega\cdot 2\) is thus
\(\omega+\omega\), while \(2\cdot\omega\) is \(2+2+2+\dots\), which is
\(\omega\). (This is the difference between the orderings 1, 2, 3,
..., [?]1, [?]2, [?]3, ... and 1, [?]1, 2,
[?]2, 3, [?]3, ....) So again, ordinal multiplication is
not commutative.



![Illustration of several ordinals. Finite ordinals are depicted by a finite set of lines. omega is depicted by a set of lines fading off into the distance. 1+omega is seen to be the same as omega, with a single infinite sequence of lines, while omega+1 consists of an infinite set of lines fading off into the distance, followed by one more. 2 times omega is an infinite set of pairs of lines, fading off into the distance, which is the same as omega.](ordinals1.png)


But for any ordinal, one can generate another ordinal by putting one
more element at the end. And for any increasing sequence of ordinals,
there is a limit of that sequence. Cantor also defined a notion of
exponentiation for ordinals, and this gives many different ordinal
numbers: 0, 1, 2, 3, ... ,\(\omega,\) \(\omega+1,\) \(\omega+2,\)
..., \(\omega+\omega(=\omega\cdot 2),\) \(\omega\cdot 2+1,\)
\(\omega\cdot 2+2, \dots,\) \(\omega\cdot 3,\) \(\omega\cdot 4,\)
..., \(\omega\cdot \omega(=\omega^2),\) ..., \(\omega^3,\)
..., \(\omega^4,\) ..., \(\omega^\omega,\)
\(\omega^{\omega+1},\) \(\omega^{\omega+2},\) ...,
\(\omega^{\omega\cdot\omega}=\omega^{\omega^2},\) ...,
\(\omega^{\omega^3},\) ..., \(\omega^{\omega^\omega},\)
\(\epsilon\_0\) (defined as the limit ordinal for the sequence
\(\omega,\) \(\omega^\omega,\) \(\omega^{\omega^\omega},\) ...).
But these ordinal numbers all correspond to the same cardinal number,
which is that of the natural numbers.



![Illustration of several ordinals, including omega times 2, omega times 3, omega squared, and omega squared times 2. omega times 2 is two copies of the illustration of omega. omega times 3 is three copies of this illustration. omega squared is an infinite set of copies of the illustration for omega, each smaller than the previous one, fading off into the distance](ordinals2.png)
  

![Illustration of two ordinals: omega squared times 3 and omega cubed. omega squared times 3 is three copies of the illustration of omega squared. omega cubed is an infinite set of copies of the omega squared illustration, each smaller than the previous one, fading off into the distance.](ordinals3.png)


At this point, one might be forgiven for thinking that there is no
cardinal number greater than that of the natural numbers, just as
there is no extended real number larger than \(+\infty\). However,
Cantor's second striking result is that the cardinality of the
positive real numbers is in fact greater than the cardinality of the
positive integers, and his third striking result is that for every
set, the set of all its subsets--its power set--has an even
greater cardinality. Although many different infinite sets of
different order types can all be put into one-to-one correspondence
with each other, there are some infinite sets that cannot. Sets whose
cardinality is equal to that of the natural numbers (like the integers
and the rationals) are said to be *countably infinite* or
*denumerable*, while infinite sets that are not countable (like
the reals, and the power set of the naturals) are said to be
*uncountable*. See proofs of the results in the



Supplement on Proofs of Theorems.



Thus, just as there is an infinite hierarchy of infinite ordinal
numbers, which Cantor represented with lowercase Greek letters, there
is also an infinite hierarchy of infinite cardinal numbers, which
Cantor represented with Hebrew letters, and in particular aleph,
"\(\aleph\)". The finite cardinals are 0, 1, 2, 3,
.... The first infinite cardinal, that of the natural numbers
(and all countable sets), is \(\aleph\_0\). Cantor's
"well-ordering principle", stating that every set can be
put into some well-ordered form (and equivalent to the Axiom of
Choice), implies that every cardinal number can be represented by an
ordinal number, and the definition of ordinal numbers ensures that for
any non-empty set of ordinal numbers there is always a first. So the
cardinals must in fact be well-ordered. So Cantor used ordinal numbers
to designate each cardinal's position in their ordering.
\(\aleph\_1\) is the first cardinal beyond \(\aleph\_0\), \(\aleph\_2\)
is the next beyond that, and after \(\aleph\_3\), \(\aleph\_4\), and so
on, we eventually reach \(\aleph\_\omega\), \(\aleph\_{\omega+1}\),
\(\aleph\_{\omega+2}\), and so on with one cardinal for every
ordinal.



Because we have this one-to-one correspondence between the cardinals
and the ordinals, one might be tempted to say that the set of
cardinals and the set of ordinals have the same order type, and then
ask what the ordinal of this order type (and its cardinality) is.
However, if there were such an ordinal, there would be a
paradox--it would have to contain, and thus be larger than, all
ordinals, including itself! This is the *Burali-Forti paradox*
(see entry
 paradoxes and contemporary logic).



Relatedly, since every set has a cardinality less than that of its
power set, there can be no set that contains everything (since such a
set would already include all its subsets, and thus be at least as
large as its power set). The two results imply that there cannot be a
set of all ordinals or a set of all sets. In a similar way one argues
that there cannot be a set of all cardinals. As a consequence of the
above results we can also answer some of the questions we raised at
the beginning: there are infinitely many cardinal numbers and
infinitely many ordinal numbers. However, there is neither a set of
all cardinal numbers nor a set of all ordinal numbers. Thus, the
infinity of the cardinals and of the ordinals cannot be measured by a
cardinal or an ordinal, for otherwise a paradox would ensue. (See also
entry on
 Russell's paradox.)



Although it took several decades from Cantor's work to find a
system of axioms for set theory that avoids these paradoxes (see the
entries on
 the early development of set theory
 and
 set theory),
 Cantor already saw, in this unreachability of the totality of all
ordinals or cardinals, a notion of "absolute infinity".
Although in his system, there are many infinite sets that are
tractable and graspable at many different levels, starting with the
natural numbers (which Aristotle had thought was merely potential
rather than actual), he discovered an even greater Aristotelian
potential infinity. This led to the distinction between a
"set" as a totality that can be grasped in the relevant
sense, and a "proper class", which is too large even for a
system that encompasses each of Cantor's many infinities.



We can define addition for cardinal numbers as the cardinality of a
set that is the union of two disjoint sets of those cardinalities. We
can define multiplication for a pair of cardinal numbers as the
cardinality of the set of all ordered pairs whose first elements are
drawn from a set of the first cardinality and whose second elements
are drawn from a set of the second cardinality. But it turns out that
these operations are relatively trivial once we get beyond the finite
cardinals--just as we saw that the sum or product of two
countable ordinals was still countable, the sum or product of two
infinite cardinals is equal to whichever of the two is larger! (At
least this operation is commutative.) Thus, one can't get to
\(\aleph\_1\) from \(\aleph\_0\) by means of addition or multiplication
(as we saw when we considered order types achieved by addition and
multiplication).



However, cardinal exponentiation is more powerful. It turns out to be
natural to define \(2^\kappa\) as the cardinality of the power set of
a set whose cardinality is \(\kappa\). (Greek letters \(\kappa\) and
\(\lambda\) are traditionally used as variables to represent infinite
cardinals, with letters \(\alpha\) and \(\beta\) used for ordinals.)
It turns out that the cardinality of the set of real numbers
(otherwise called the "continuum" for its role in
representing continuous space) is the same as that of the power set of
the natural numbers, \(2^{\aleph\_0}\), and Cantor showed that
\(2^{\aleph\_0}>\aleph\_0\). Cantor conjectured that \(2^{\aleph\_0}\)
was in fact equal to \(\aleph\_1\), and this conjecture was called the
"Continuum hypothesis". (See the entry on
 the continuum hypothesis
 for more on this conjecture and why it hasn't been
settled.)



Just as \(\aleph\_1\) is a cardinality that can't be reached by
the operations of ordinal addition, multiplication, or exponentiation,
even in the limit, one might conjecture that even before one reaches
the Cantorian absolute infinity, there are further cardinalities that
can still be grasped in some sense, but can't be reached even by
the stronger operation of cardinal exponentiation, even in the limit.
Such conjectures have turned out to be surprisingly fruitful for the
study of set theory and mathematical logic generally. (See the entry
on
 large cardinals and determinacy.)



Although we have defined addition and multiplication for both ordinals
and cardinals, their features make it hard to make sense of
subtraction or division. First, the non-commutativity of the ordinal
operations, and the triviality of the cardinal operations, makes it
hard to define meaningful inverses of these operations. (If it's
possible to find an ordinal or cardinal that can be added or
multiplied to a first one to get a second one, then it is usually
possible to find infinitely many that can.) But more importantly, the
conceptual ideas of counting (whether by order type or by bijection)
don't really allow for negatives or fractions, the way that the
conceptual ideas of measuring and geometry do (considered in section
 3.2).
 Counting involves treating each element as discrete and unique, and
there is no way for multiple elements to combine to yield nothing (as
subtraction requires) or to yield a unit (as division does). However,
measurement (e.g., of distance) involves a sort of structure on the
thing being measured so that some measurements can be fractions of
others, or can point in the opposite direction, which yields
meaningful notions of division and subtraction.



This section has given just an introductory taste of the mathematics
of ordinal and cardinal numbers--it has come to be called
*Cantor's Paradise*. But we have already seen many
characteristic features of these notions of infinity that
differentiate them starkly from those discussed in the previous
section. Just as the differences between the natural numbers and the
real numbers demonstrate the differences between counting elements of
sets, and measuring lengths, areas, and so on, the differences between
Cantor's cardinals and the extended reals further demonstrate
differences between counting and measuring.



For further discussion of the kind of material presented in this
section, see the SEP entries on:
 set theory,
 the early development of set theory, and
 the axiom of choice.
 An excellent introduction to basic set theory is Enderton (1977); an
informal but still rigorous introduction is Sheppard (2014); more
advanced texts include Devlin (1993), Kunen (1983), and Jech (2006).
For the higher reaches, to which we will come back in section
 4,
 see Kanamori (2003).



Set theory provides a theory of cardinality that implements the idea
that "sameness of size" upholds the bijection intuition.
The recent theory of numerosities, developed by Benci, Di Nasso, and
Forti, by contrast, upholds the part-whole intuition. See the



Supplement on Theories of Numerosities.


### 3.4 Infinitesimals and hyperreals



We have already discussed a notion of \(+\infty\) and of \(-\infty\)
that is designed to provide values for real-valued functions to take
as limits at special points. But the understanding of limits
themselves was originally thought to require a notion of
"infinitesimally small" distances. While these quantities
were considered problematic for several centuries, in recent decades
some mathematical entities with their properties have been rigorously
studied.


#### 3.4.1 Newton and Leibniz



An infinitesimal is a number smaller in absolute magnitude than any
positive finite number, and yet not zero. Infinitesimals have had a
chequered history. Early work in the calculus, as we have seen when
presenting the structure of L'Hopital's 1696
treatise, was mostly based on a geometrical or kinematic (i.e. based
on motion) understanding of infinitesimals. Here we treat
infinitesimals arithmetically, i.e. within the context of the real
number system, while conveying the key features of how the early
analysts made use of them. To figure out the slope of the function
\(f(x)=x^2\) precisely at one point, one considered an
"infinitesimally small number" \(\epsilon\) and considered
the slope of the straight line through \(f(x)\) and
\(f(x+\epsilon)\).



![Illustration of a curve y=f(x) and a tangent line to the curve, together with an 'infinitesimal' zoom in on the intersection, showing that the tangent line can be seen as a line connecting the point (x,f(x)) to the infinitesimally close point (x+epsilon,f(x+epsilon)), so the slope of the tangent line is (f(x+epsilon)-f(x))/epsilon.](slope.png)

To find the slope of a function, one considers the slope of a straight
line that intersects it at two "infinitesmally close"
points.




This slope is equal to \(\frac{(x+\epsilon)^2-x^2}{\epsilon}\). In
order for this fraction to make sense, \(\epsilon\) must be non-zero.
However, we can calculate that this value is
\(\frac{2x\epsilon+\epsilon^2}{\epsilon}\), or \(2x+\epsilon\). At
this point, we no longer need \(\epsilon\) to be non-zero, so the
slope can be said to be just \(2x\). This sort of slippage between
non-zero and zero for these infinitesimals is what made Berkeley refer
to them as "the ghosts of departed quantities". However,
engineers, scientists and mathematicians who actually made use of the
calculus rested content in the knowledge that the calculus delivered
the goods.



In the 19th century, Cauchy, Bolzano, Weierstrass, Dedekind and Cantor
sought to establish foundations for real analysis that gave no role to
infinitesimals: in what became the canonical account of the calculus,
Cantorian set theory and the \(\epsilon\)-\(\delta\) formalization of
the notion of a limit described in section
 3.2
 allowed for a fully rigorous development of real analysis.



Instead of taking *particular infinitesimal* values, one
*quantifies* over values of *real-valued variables*
\(\epsilon\) and \(\delta\). For example, the claim that the slope of
the squaring function at \(x\) is \(2x\) is interpreted as saying that
*for any* desired degree \(\epsilon\) of approximation,
*there is* some finite \(\delta\) such that for any \(x'\)
within \(\delta\) of \(x\), the slope of the line from \((x,x^2)\) to
\((x', x'^2)\) approximates \(2x\) to within \(\epsilon\). A single
infinitesimal is replaced by a relation involving two nested
quantifiers. The process of elimination of infinitesimal quantities
from the calculus was a central part of a larger process known as the
"arithmetization of analysis", which aimed at removing
kinematical and geometrical notions from the calculus in favour of
purely arithmetical notions. (These are broadly construed to include
the arithmetic of real and complex numbers. For a recent account of
the history of real and complex analysis in the 19th century that also
pays attention to foundational issues see Gray (2015).)



This infinitesimal-free program accomplished its goals
successfully--among its major accomplishments were the rigorous
definition of a continuous function at a point and the definition of
the Riemann integral. However, one should keep in mind that other
areas of mathematics, such as geometry, continued exploiting
infinitesimal considerations and studied extensively
*non-Archimedean* number systems. Archimedes' axiom
states:



> 
> 
> given any two areas, or two distances, or any two quantities of the
> same sort, say \(A\) and \(B\), it is possible to add \(A\) to itself
> finitely many times so that the quantity obtained is greater than
> \(B\).
> 
> 
> 



Non-Archimedean systems are ones in which this axiom does not hold. If
the 17th century engagement with infinity, and Cantor's work in
set theory, can be seen as revolutions, the study of non-Archimedean
mathematics in the second half of the 19th century can be likened to
an infinitary uprising.



Many of the quantities considered in 17th century calculus, such as
Leibnizian infinitesimals, do not obey the Archimedean axiom. An
infinitesimal can be added to itself any finite number of times but
the outcome of that process will never be greater than any finite
quantity, however small. A pervasive historiographical tradition has
argued that with the elimination of infinitely small quantities from
the calculus, non-Archimedean quantities were relegated to engineering
practice for a long time. According to the standard account, it was
only in the 1960s that infinitesimals came back when Abraham Robinson
presented his theory of non-standard analysis, which has received a
lot of attention from philosophers and mathematicians (see section
 3.4.2).
 Robinson's theory gave a legitimate mathematical status to
infinitely small and infinitely large quantities in the reconstruction
of the infinitesimal calculus, now developed accordingly to rigorous
model-theoretic techniques. For a long time, Robinson's work was
hailed as the first successful effort to develop a system of
non-Archimedean quantities. But Philip Ehrlich, in a series of
fundamental papers (including 1994, 2012), has argued that this
widespread perception needs serious questioning. Indeed, he has shown
convincingly that interest in non-Archimedean mathematics emerged in
the 1870s and continued to grow in the hands of mathematicians such as
Veronese, du Bois-Reymond, Levi-Civita, Hahn, Stolz, Hardy, and
others.



It would be out of place in this entry to attempt even a small survey
of the developments mentioned above. We simply refer to the reader to
Ehrlich (1994) and (2006). The interconnection with many important
related issues such as Conway's surreal numbers (see section
 3.4.3)
 and other alternative approaches to the construction of the real
numbers, such as smooth infinitesimal analysis mentioned at the end of
section
 3.4.2,
 cannot be properly addressed here. See Salanskis and Sinaceur (1992),
Ehrlich (1994), Berger, Osswald, and Schuster (2001), and Ehrlich
(2012).


#### 3.4.2 Non-standard analysis and infinitesimal analysis



As a result of the rigorous definitions in the calculus mentioned
above, from the mid-19th century most mathematicians working in
analysis abandoned infinitesimals. However, in the mid-20th century,
Robinson (1966) showed that it is possible to give a rigorous
definition of infinitesimals, and that infinitesimals can be used in a
non-standard development of real analysis (D. Laugwitz also did
similar work around the same time, but Robinson's system has
been more widely discussed). While he developed his non-standard
analysis using model theory, subsequent developments have also been
grounded in algebra and topology. Robinson's approach supplies
an extended number system--the *hyperreal* number
system--that contains the standard real number system, plus
further 'infinitesimal' numbers whose absolute values are
greater than 0, but less than any positive standard real number.
Robinson's construction of the hyperreals provides a set with
the same cardinality as the standard reals. Simple modifications of
the construction can create sets of hyperreals of larger
cardinality.



Importantly, due to the logical techniques used in its construction,
Robinson's system behaves exactly like the standard finite real
numbers for any sentence expressible within the algebraic language of
addition and multiplication. Thus, every number other than 0 has a
multiplicative inverse, and if \(x > y\), then \(1/y > 1/x\). In
particular, this means that if \(\epsilon\) is a positive
infinitesimal number, then \(1/\epsilon\) is an infinitely large
number! Unlike Cantorian infinities of counting from section
 3.3
 these infinitely large numbers are subject to subtraction and
division as well as addition and multiplication, and unlike the
infinities of the extended real line from section
 3.2,
 they behave just as nicely as the finite numbers with respect to
them. For instance, statements such as these hold of standard real
numbers as well as of the new infinitely small and infinitely large
numbers:



\(x+y=y+x\) (commutativity of addition);
  

\(x\cdot y=y\cdot x\) (commutativity of multiplication);
  

\(x(y+z)=xy+xz\) (distributivity of multiplication over addition).



In fact, Robinson's hyperreals satisfy a "transfer
principle"--if statements are formulated entirely within a
first-order language for the reals, then they are true of the standard
reals if and only if they are true for the hyperreals. So any proof of
such a theorem in one system can be transferred to the other. This
sometimes greatly simplifies calculations and proofs of theorems.



Consider the limit of the quantity \(((x+h)^3-x^3)/h\) as \(h\)
approaches 0. In the standard reals, to show that this is \(3x^2\), we
need to show that for every \(\epsilon\) there is a \(\delta\) such
that for any value of \(h\) less than \(\delta\), the corresponding
value of this function is less than \(\epsilon\) away from \(3x^2\).
In this case, it turns out that choosing \(\delta < \epsilon/4x\)
works when \(x\) is sufficiently large, and choosing \(\delta <
\epsilon\) works when \(x\) is sufficiently small, but figuring out
these choices is difficult.



For the hyperreals though, it is sufficient to show that this value is
infinitesimally close to \(3x^2\) whenever h is infinitesimal.



\[\begin{align}
\frac{(x+h)^3-x^3}{h} &= \frac{x^3+3x^2h+3xh^2+h^3-x^3}{h} \\
 &= 3x^2+3xh+h^2, 
\end{align}\]


and for any real \(x\), \(3xh+h^2\) is infinitesimal whenever \(h\) is
infinitesimal.) For any particular real \(\epsilon\), this shows that
there is some *hyperreal* \(\delta\) that works (namely, any
infinitesimal), and by the transfer principle we can conclude that for
this real \(\epsilon\) there is some *real* \(\delta\) that
works, and we no longer need to worry about the details of how to find
it. Thus, we can validate the reasoning of Newton and Leibniz that
allows them to treat infinitesimals as non-zero for calculations until
we get to the final result, and then treat them as zero at the end.
They really work like the "ghosts of departed quantities"
that Berkeley satirized! (The extent to which Robinson's system
is a vindication of Leibniz and Newton was the subject of extended
discussion in articles by Robinson and others. See Bos (1974) for a
classic source of this debate.)



For results stated in a first-order logical language, the hyperreals
and the standard reals satisfy the transfer principle. But for results
about *sets*, they behave differently. Every bounded set of
standard reals has a least upper bound. However, for instance, the set
of infinitesimal hyperreals is bounded (every member is less than
.00001, among other bounds), but there is no *least* upper
bound (no infinitesimal is an upper bound for all of the others, and
every finitely large upper bound can be decreased by some
infinitesimal amount to give a smaller one). Edward Nelson (1977)
pioneered an alternative approach--Internal Set Theory--on
which the basic language of mathematics is enriched in order to allow
us to distinguish between standard and non-standard real numbers, as
well as "internal" and "external" sets. On
Nelson's approach, infinitesimals are non-standard real numbers
that are smaller in absolute value than any positive standard real
numbers. "Internal sets" are those that can be defined in
the basic language, and they behave just the same as standard sets of
standard reals--for instance, bounded internal sets always have a
least upper bound. But the set of all infinitesimals, just like the
set of all standard real numbers, is an "external set" of
the theory that can't be defined within the language, and thus
doesn't necessarily have a least upper bound.



The approaches pioneered by Robinson and Nelson do not allow us to
prove results about the standard real numbers that *cannot* be
proved using standard real analysis. However, these approaches do
provide simpler--and, in some sense, more intuitive--proofs
of many theorems of standard real analysis. (On the pedagogical
benefits of non-standard analysis, see, for example, Keisler (1976)).
And there are cases of results in real analysis that were first proven
using non-standard real analysis (see, for example, Bernstein and
Robinson (1966).) Moreover, these approaches clearly show that we do
not need to adopt the \(\epsilon\)-\(\delta\) formalization of the
notion of a limit in order to have access to a fully rigorous
development of real analysis.



The literature on non-standard analysis is very rich. See Dauben
(1995) for a biography of Robinson with special emphasis on
non-standard analysis. See also Goldblatt (1998) for a recent formal
introduction and Cutland, di Nasso, and Ross (2006) for recent
mathematical developments. The reader is referred to the extensive
bibliographies contained in those volumes for further references.



An interesting alternative to nonstandard analysis, which allows for a
development of substantive parts of mathematics, goes under the name
of *(smooth) infinitesimal analysis*. This differs from both
ordinary and nonstandard analysis by allowing *nilpotent
infinitesimals*, namely 'linelets' \(dx\) such that
\(dx\neq 0\) but \(dx\cdot dx = 0\). The consistency of such a theory
is proved using toposes in category theory. The best exposition of the
topic is Bell (1998b) (see also Bell 1988a, 2019); philosophical
aspects of the theory are discussed in Hellman and Shapiro (2018).
Arthur (2013) discusses infinitesimal analysis in connection to
Leibniz and makes points similar to those made by Bos (1974) on
Leibniz and non-standard analysis. Constructive interpretations of
non-standard analysis are developed in Salanskis (1999), which
includes discussions of Nelson's approach as well as the French
school of non-standard analysis (Reeb, Harthong). For further
discussion of infinitesimals, see Davis (1977), Thomason (1982), Bell
(2005), and the entry on
 continuity and infinitesimals.


#### 3.4.3 Surreal numbers



Dedekind showed how to fill the gaps *between* rational
numbers; Cantor showed how to extend (ordinal and cardinal) numbers
*beyond* the existing finite numbers. John Conway (1976)
integrated both ideas. He developed a very different system that
generalizes von Neumann's representation of Cantor's
ordinals, as well as Dedekind's representation of the real
numbers, to generate a much larger field that has become known as the
"surreal numbers". It contains a copy of each ordinal and
cardinal number, while defining operations that work just like
addition, subtraction, multiplication, division, exponentiation, and
the taking of roots, on the standard reals. In particular, even the
infinite and infinitesimal surreal numbers are amenable to these
operations. Thus, as well as familiar numbers, we now have numbers
such as \(\sqrt\omega, \omega/2, -\omega, 1/\omega, -\omega^\omega,\)
and so on. Indeed, as Ehrlich (2001, 2012) observes, the surreal
numbers may plausibly be regarded as including "all numbers
great and small"! The surreal numbers can apparently be applied
in cases where there is no straightforward way to use hyperreals, as
for example in the treatment of Pascal's Wager, discussed in
section
 7.3--see
 Hajek (2003a).



Because the field of surreal numbers contains copies of all the
ordinals, it is too big to form a set. But because the operations
behave like the operations on the standard reals, these copies of the
ordinals don't represent counting. See the



Supplement on the Construction of Surreal Numbers




for a summary of Conway's construction of them. Other
constructions of this same structure have been carried out by Knuth
(1974) and Gonshor (1986) in more introductory texts.


### 3.5 Wrapup



Let us take stock. In response to worries that infinities in
mathematics are suspect (section
 2),
 rigorous mathematical theories of infinity have been developed (this
section). But one might worry that, even if we can talk with
mathematical rigor about infinities, they do not correspond to or
apply to anything in the real world (as we think finite quantities do,
however they do). Infinities might just be castles in the sky.
Furthermore, one might suspect that we can, by some further
mathematical developments, remove any reference to infinities in any
practically important mathematics. The following section places this
dialectic in the context of general questions about mathematical
ontology, canvassing some important historical attempts to expunge the
infinite from mathematics. It then explains the very
difficult--perhaps insurmountable--challenges that any such
attempt faces.


## 4. Mathematical ontology



Various questions about infinity naturally arise in the course of
theorizing about ontology. If mathematical objects exist, are there
infinitely many of them? Do individual infinite objects like the ones
mentioned above exist, in addition to the infinitely many individual
finite numbers? This article will not directly discuss the question of
whether and in what sense mathematical objects exist. Instead, we will
focus on the question of whether the infinities discussed above exist
in the same sense as the finite integers. For more on the general
questions of mathematical existence, see the entries on:
 logic and ontology,
 philosophy of mathematics,
 platonism in the philosophy of mathematics,
 nominalism in the philosophy of mathematics,
 fictionalism in the philosophy of mathematics,
 naturalism in the philosophy of mathematics, and
 logicism and neologicism.



Most viewpoints in the philosophy of mathematics accept the existence
of all of the finite and infinite objects mentioned so far in exactly
the same way that they accept the existence of finite integers.
(Platonists might accept that this is literal existence, while
fictionalists accept this as some sort of fictional existence, and
others might have a different idea of what this means.) Standard set
theories can prove the existence of all these objects, and for most
mathematicians and philosophers, this is all that is needed. Logicist
and neologicist accounts of mathematics may obtain the existence of
infinite sets or infinitely many numbers by explicit postulation (as
in the case of the axiom of infinity in Whitehead and Russell's
*Principia Mathematica*) or as the outcome of an implicit
postulation (such as Hume's Principle in Scottish neologicism,
see Hale and Wright 2001, Heck 1997, 2011). While the axiom of
infinity is easily stated and understood, Hume's Principle has a
peculiar form, for it postulates the existence of a function \(\#\)
that sends concepts into objects while respecting an equivalence
relation \(\approx\) among concepts. Formally it is stated as
follows:

 
\[\tag{HP} \forall B\forall C\, \# B = \# C \text{ iff } B \approx C \]


where \(B \approx C\) is short-hand for one of the many equivalent
formulas of pure second order logic expressing the equivalence
relation "there is a one to one correlation between the objects
falling under \(B\) and those falling under \(C\)". Informally,
it can be read as saying that two concepts \(B\) and \(C\) have the
same 'number' if and only if there is a one to one
correspondence among the objects that fall under \(B\) and those that
fall under \(C\). Principles like HP that define a function from an
equivalence relation are called *abstraction* principles. By
presupposing the existence of a function that sends concepts into
objects, Hume's Principle exploits the possibility of defining
infinitely many concepts that do not stand in the equivalence relation
mentioned in its right-hand side to generate infinitely many natural
numbers. There are other varieties of neologicism that do not
postulate Hume's Principle or an axiom of infinity at the outset
but yield infinitely many natural numbers by means of other logical
principles (see for instance Linsky and Zalta 2006). In addition, all
these varieties of neologicism generate at least an infinite cardinal
numbers and what is of philosophical relevance here is the different
resources they use to establish these results.



Incidentally, Frege's logicism and the neologicist program use
one-to-one correspondence to state identity criteria for
"concepts" even when infinitely many objects fall under
the concepts. For alternative criteria for assigning numbers to
infinite concepts see Mancosu (2015) and (2016).



In light of the paradoxes for early set theories (Russell's
paradox, the Burali-Forti paradox, and others), some mathematicians
and philosophers worried that standard set theory might be
inconsistent as well. One alternative viewpoint on mathematics is
*intuitionism*, which only accepts the existence of
mathematical objects whose construction can be carried out in some
sense by the human mind. Intuitionism requires a revision of logic,
since this limitation invalidates the Law of Excluded
Middle--there are cases in which we can prove that the
*non*-existence of a certain type of object would lead to a
contradiction, but don't have any method of constructing such an
object, so that there might be a truth-value gap. Intuitionists often
accept the Aristotelian limitation to "potential
infinities", rather than "actual infinities", but
there is also sophisticated intuitionistic reasoning about what types
of infinite entities might exist. (For more, see the entries on
 intuitionistic logic
 and
 intuitionism in the philosophy of mathematics.)



Another viewpoint, associated with David Hilbert, is called
*finitism* (see Hilbert 1926). Most finitists accept classical
logic, but worry about the consistency of theories of infinite
objects. Hilbert's worries about consistency were fueled by the
paradoxes that the new infinitary set-theoretic mathematics was giving
rise to (Cantor's inconsistent sets; Burali-Forti Paradox;
Russell's paradox etc.) Hilbert was convinced that
quantification over such infinite totalities was at the root of the
troubles. Finite objects, like configurations of strokes corresponding
to the natural numbers, and finite sentences of a formal language, are
taken by the Hilbertian finitist to be unproblematic, since these
objects can in some sense be grasped individually and thus in their
(potential) totality. But infinite objects are taken to be
problematic: this includes Cantor's higher ordinals and
cardinals, and all the geometric, algebraic, and topological objects
of which mathematicians were starting to develop detailed theories at
the turn of the 20th century.



Hilbert's proposed project (sometimes taken to be the starting
point for *formalism in the philosophy of mathematics*) was to
replace talk of these infinitary entities themselves with talk of the
finitely long sentences that we ordinarily interpret as being about
the entities. His goal was to axiomatize the theories of these
infinite objects, and then to prove, using finitary means of syntactic
reasoning about the language, that these theories are consistent.
While this idea doesn't *deny* the existence of the
infinite objects, it suggests a methodological approach of only
literally accepting the finitary objects, whether strokes standing for
the integers or sentences. (See the entry on
 Hilbert's program.)



Some mathematicians and philosophers have adopted finitism not merely
as a methodological viewpoint, but also as a metaphysical one. Finite
objects, like numbers and sentences, exist (in whatever sense
mathematical objects exist), but infinite objects (like the completed
set of all the natural numbers, or even arbitrary irrational numbers
represented by Dedekind cuts) don't. Versions of this view are
often attributed to the 19th century number theorist and algebraist,
Leopold Kronecker, who is quoted as saying "The dear God created
the whole numbers; everything else is the work of man."
Kronecker criticized Cantor's work as theology rather than
mathematics. Hilbert started his program with the intent of defending
Cantor while working within a framework dialectically acceptable to
Kronecker's allies. But when Kurt Godel proved that no
finitary theory for arithmetic and syntax can even prove its
*own* consistency, let alone prove the consistency of a
stronger theory for talking about completed infinities,
Hilbert's program was taken to have failed at disabusing the
metaphysical finitists. Godel's incompleteness theorems
apply most notably to Peano Arithmetic. The language of Peano
Arithmetic is given by \(\{0,\,',\, +,\,\times\}\), where *'*
is the successor function (it adds 1 to each number).
Within it one can express ordinary arithmetical claims such as the
commutativity of addition and the infinitude of prime numbers. The
axioms of Peano Arithmetic tell us that the function *'*
is one-one; that 0 is not the successor of any
number; that \(+\) and \(\times\) satisfy the usual recursive
definitions; and finally we have a schema of induction for every
formula \(A(x)\) expressible in the language, i.e. if \(A(0)\) and for
all \(x, A(x) \rightarrow A(x')\), then for all \(x, A(x)\).



The detailed foundational work carried out in set theory and other
foundational areas has in many ways dispelled the fear of impending
doom that characterized the reaction to the paradoxes at the beginning
of the twentieth century. As a consequence, most mathematicians today
are perfectly happy to work with completed infinities. But there are
still some finitists and intuitionists.



An intermediate position is that defended by classical
"predicativists" such as Poincare and Weyl. The
theory, presented in a satisfactory logical way by Feferman and
others, accepts the excluded middle on the natural numbers (and as
such it is arguably committed to the existence of the set of natural
numbers and in any case to accepting bivalence on the natural numbers)
but does not accept the existence of the power set of the natural
numbers. According to *predicativism* (see Feferman 2005), sets
exist only if they are definable in some non-circular linguistic way.
In accepting the excluded middle on the natural numbers and in making
the existence of sets depend on our definitional abilities, this
position is a sort of a compromise between a classical and a
constructive viewpoint. In 1918, Hermann Weyl (Weyl 1918; see Kaufmann
1930 for a related program) presented the foundations of analysis
within this framework and showed that a great part of classical
analysis can be carried out within the framework by replacing talk of
arbitrary sets of real numbers with arithmetical sequences of real
numbers. Feferman 1988 gave a detailed formal presentation of the
theory and proved that, on a certain reconstruction, the theory is a
conservative extension of Peano Arithmetic. In addition, he also used
the theory to state an important conjecture concerning how much
mathematics is needed in physics. In Feferman 1984 and 1987, he
proposed that all of mathematics used in physical theory can be
recaptured in a predicative system of analysis. Using the
metatheoretical result of conservativity mentioned above, he also
exploited the argument to claim that Quine and Putnam's
indispensability arguments (see the entry on
 indispensability arguments in philosophy of mathematics)
 at best commit us to what Peano Arithmetic commits us to.



In contrast to the possibility of eliminating infinity as just
described stand a number of results that show that some finitary
statements can only be proved through infinitary considerations. These
results originally emerge with Godel's incompleteness
theorems (Godel 1931) but have been recently refined by
displaying statements of mathematical interest (whereas
Godel's statements are of metamathematical interest but
have no obvious mathematical interest). In order to understand the
conceptual distinctions required, let us grant --as most
logicians do--that all finitistic modes of reasoning are included
in first-order Peano Arithmetic (henceforth PA).



A consequence of Godel's incompleteness theorems is that
under the assumption that PA is consistent one can find a finitistic
statement such that neither it nor its negation is provable from Peano
Arithmetic. Godel showed, through subtle coding of
metamathematical notions in the language of arithmetic, how to express
in the language of arithmetic a formula \(G\) that "says"
of itself that it is not provable. One can also ascertain that the
formula is true on the natural numbers. Since all the finitary
reasoning is assumed to be included in PA, establishing the truth of
the Godel sentence and of the new incompleteness results requires
appeal to some "infinitary" principles (when the truth of
the Godel sentence \(G\) is established through appeal to the
statement expressing the consistency of PA, it is establishing the
latter that requires some portion of infinitary reasoning, such as
induction up to an infinite ordinal called \(\varepsilon\_0)\).



The situation is the same for the statement Con(PA) expressing the
consistency of PA. Godel's second incompleteness theorem
shows that neither it nor its negation can be proved from PA but an
appeal to some infinitary reasoning shows it to hold in the natural
numbers. While perfectly fine for the logician's need and
central to the evaluation of Hilbert's program,
Godel's sentences appear concocted from the point of view
of the practicing mathematician. Within Hilbert's program
statements of PA expressible either without quantifiers or with a
string of universal quantifiers followed by a non-quantificational
formula count as finitistic statements. The statements \(G\) and
Con(PA) mentioned above also belong to this class. Finitistic
statements with obvious mathematical relevance include basic
properties such as the commutativity of addition as well as the
statement of Fermat's last theorem (whose proof has been
established using higher mathematics but logicians are convinced that
it can also be carried out in PA). Logicians have not been able to
find finitistic statements with obvious mathematical significance that
require a detour through the infinite, but they have been able to do
the next best thing. They have found statements that have the form
\((\forall x) (\exists y) A(x, y),\) which express a certain
functional connection between numbers and have shown that such
statements, although true, cannot be proved using only the resources
of PA. Among the most famous such results are a modification of
Ramsey's finite theorem provided by Paris and Harrington (1977),
and the proof that a theorem by Goodstein (1944) cannot be proved in
PA (Kirby and Paris 1982). There are stronger results that are
independent of even stronger systems that are studied within the
context of reverse mathematics (e.g., Kruskal's theorem is
independent of predicative analysis--see Simpson 1985, 2002).



Such results show that even an arithmetical theory such as PA can
express statements of mathematical significance (as opposed to
statements concocted for logical purposes) that require some detour
through the infinite to be proved, even though they can be stated
purely arithmetically. In contrast to arithmetic, the mathematical
incompleteness of set theory was shown by Godel and Cohen for
important statements such as the Axiom of Choice, the continuum
hypothesis, etc. It is important to emphasize here that logicians
working in set theory, recursion theory and proof theory probe the
mysterious role of the infinite in proving results about the finite.
It could be said that set theorists are mainly concerned with
understanding how the demonstrable mathematical incompletability of
Zermelo-Fraenkel (with Choice, i.e. ZFC) set theory, which is a
consequence of results by Godel and Cohen, can be addressed by
finding new principles that will allow us to solve some of the most
pressing questions concerned with the structure of the real numbers.
In other words, since ZFC cannot be taken as a sufficient basis for
the mathematics of infinity much of contemporary set theory is trying
to solve the problem by finding new principles, which often take the
form of assuming the existence of very large cardinals (see the entry
on
 independence and large cardinals).
 The hope is that this work will lead to settling the Continuum
Hypothesis and other major problems about the projective sets (on
projective sets see entry on
 set theory).



Recursion theorists are also trying to understand the role that
infinitary principles or compactness arguments play in our
determination of results about the finite. And proof theorists would
like to know when certain infinitistic theories can be justified
through finitary means. Obviously, a more precise description of these
developments goes well beyond the technical knowledge that we can
presuppose here.



Most working mathematicians don't worry about the existence of
infinitely large sets and other objects. There are some other
ontological worries about particular infinite sets, related to the
Axiom of Choice, and some of the larger cardinalities mentioned above
in the section on Cantor. But bigger worries arise in the context of
whether there can be physical infinities.



For collections of sources on the classical foundational positions
(finitism, intuitionism, predicativism) see van Heijenoort (1967),
Ewald (1996), and Mancosu (1998). On finitism and intuitionism see the
entries
 Hilbert's program
 and
 intuitionism in mathematics.
 On predicativity see Feferman (2005). On Paris-Harrington see Katz
and Reimann 2018; on Goodstein's theorem see the friendly
presentation in Stillwell (2010). Stillwell (2010) also has a chapter
on large cardinals; for recent directions see Woodin (2011) and Steel
(2015). On the interplay between finite and infinite in recursion
theory see Hirschfeldt (2015).


## 5. Paradise lost? Paradoxes and puzzles involving infinity



The latter part of this entry will explore selected applications of
mathematical concepts of infinity in theories of probability,
decision, and spacetime, and some associated paradoxes. Before we turn
to those theories, we warm up with some paradoxes and puzzles that
link mathematics, metaphysical possibility, and physical possibility.
There are many different paradoxes and puzzles that we might have
included in this section. We consider a small sample of paradoxes and
puzzles that some--e.g. Pruss (2018a)--have thought might
motivate a return to Aristotle's views on the impossibility of
actual infinites.



In the



Supplement on al-Ghazali's objection,



we discuss a puzzle due to al-Ghazali that is of historical
interest. For more see, for example, Rucker (1982), Moore (1990/2019),
Oppy (2006), and Huemer (2016).


### 5.1 Hilbert's Hotel



Hilbert's Hotel has infinitely many rooms, labelled 1, 2, 3,
..., each of which is currently occupied by a guest. Despite the
fact that the hotel is already full, a new guest who turns up at
reception is readily accommodated: for each n, the guest in room \(n\)
is moved to room \(n+1\), and the new guest is installed in room 1.
Indeed, despite the fact that the hotel is already full, it can
accommodate infinitely many new guests: for each \(n\), the guest in
room \(n\) is moved to room \(2n\), and the new guests are installed
in the odd-numbered rooms. Of course, if the infinitely many people in
the odd-numbered rooms check out, there are infinitely many people
left in Hilbert's Hotel; but if infinitely many people check out
from all but the first three rooms, only three people remain.



Some philosophers have thought that Hilbert's Hotel supports an
argument against the possibility of physically realized
infinities:


1. If there could be physically realized infinities, then there could
be a hotel with infinitely many rooms.
2. But if there could be a hotel with infinitely many rooms, then the
events described in the preceding paragraph could occur.
3. But it is absurd to suppose that the events described in the
preceding paragraph could occur.



So there cannot be physically realized infinities. (See, e.g., Craig
(1979).)



This argument faces various challenges, depending on one's views
about what physical possibility amounts to. The first premise may be
challenged: perhaps some kinds of physical infinities can be realized
even though other kinds of physical infinities cannot: for example,
perhaps there can be infinitely many stars even though there cannot be
a hotel with infinitely many rooms. The second premise may also be
challenged: even if there could be a hotel with infinitely many rooms,
perhaps the events described in the story could not occur--the
story was told at a high level of abstraction, and the details may
matter. And the third premise is also questionable: it is not clearly
absurd to suppose that there could be an infinite hotel in which
guests come and go in the manner described.



For further discussion of Hilbert's Hotel, see Gamow (1946),
Huby (1971), Rucker (1982), Moore (1990/2019), Oppy (2006), Kragh
(2014), Huemer (2016), and the entries on
 supertasks
 and
 cosmology and theology.


### 5.2 Thomson's Lamp



Suppose that we have a lamp and a means of turning the lamp off and
on. Suppose that the lamp is initially off. In the first minute, we
change the state of the lamp from off to on. In the next half minute,
we change the state of the lamp from on to off. ... In the next
\(1/2^n\) minute we change the state of the lamp to the other state
... . The question that we are invited to answer is: what is the
state of the lamp at the end of the second minute?



The scenario is under-described. We can imagine that the means of
turning the lamp off and on requires a spacetime location at which at
least one physical quantity is infinite. If so, it is plausible to say
that the case is impossible: there could not be such a lamp, and so
there is no question to answer. Suppose, for example, that there is a
switch that moves the same distance back and forth to turn the lamp on
and off. Consider the speed at which the tip of the switch is
travelling at the end of the second minute.



We can also imagine that the means of turning the lamp off and on does
not involve any spacetime location at which at least one physical
quantity is infinite; Grunbaum (1968) describes a scenario that
fits this specification. In that case, the means of turning the lamp
off and on converges to a specified state at the end of the two
minutes, and there is an answer to the question that lies in the
details of the specified state. But that answer is underdetermined by
the brief description that we were initially given, as Benacerraf
(1962) argues: we can have the lamp on at the end of the two minutes,
or off at the end of the two minutes, depending upon the details of
the implementation of Grunbaum's proposal. Huemer (2016:
198-201) points out that if we hold fixed enough physics, then,
before the end of the two minutes, the activation of the mechanism
will stop changing the state of the lamp. So, depending upon your
views about the range of what is possible, you may regard as
impossible even cases in which there is no spacetime location at which
at least one physical quantity is infinite.



Thomson's lamp is an example of a *supertask* (Thomson
coined this term): a process that involves infinitely many steps
completed in a finite amount of time. The trick is that the steps are
completed in shorter and shorter periods of time, corresponding to a
convergent series. The lamp is one of many examples of supertasks that
various authors have found paradoxical, while other authors have been
less troubled by them. See the entry on
 supertasks.



For further discussion of Thomson's lamp, see: Thomson (1954,
1967), Benacerraf (1962), Chihara (1965), Grunbaum (1968, 1973),
Craig (1979), Berresford (1981), Moore (1990/2019), Earman and Norton
(1996), McLaughlin (1998), Oppy (2006), Huemer (2016), and Pruss
(2018a).


### 5.3 Paralysis



Suppose that Achilles wants to run from \(A\) to \(B\) but there are
infinitely many gods who--unbeknownst to one another and to
Achilles--each have reason to stop him from getting to \(B\). God
1 resolves to instantaneously paralyse Achilles if and when he reaches
halfway between \(A\) and \(B\). God 2 resolves to instantaneously
paralyse Achilles if and when he reaches one quarter of the way from
\(A\) to \(B\). ... God \(n\) resolves to instantaneously
paralyse Achilles if and when he reaches \(1/2^n\) of the way from
\(A\) to \(B\). ... Since all of the Gods are able to act on
their resolve, Achilles is unable to move: for, if he moved, he would
violate the intentions of infinitely many Gods. But, until he does
move, none of the Gods act on their intentions. So what actually stops
him from moving? Isn't it absurd to suppose that someone can be
rendered immobile by a nested sequence of conditional intentions upon
which no one acts?



Suppose instead that each of the Gods erects a force field, placed in
a parallel manner to the previous case, that it is impossible for
Achilles to cross. Then Achilles is completely immobilized. On the
assumption that it is possible for an infinite number of Gods to
collectively create such a force field in the manner described, there
is a straightforward explanation for Achilles' inability to
move. Of course, granted this assumption, there is no single God whose
force field immobilizes Achilles; indeed, there is no finite
collection of Gods whose collective force field does so; indeed, no
force field touches him at all. Is this possible? Presumably one
should come to the same verdict regarding the conditional intentions
case as in this case.



Depending on your views about the range of possibility, there is much
in this story that you may think is impossible. You may think that it
is impossible for there to be Gods who can act as required; for
example, depending on your conception of the Gods and their actions,
you might think that the story requires instantaneous action at a
distance. You may think that it is impossible for force fields to be
positioned with unbounded degrees of accuracy. And so on. However, if
there is nothing in the set-up that makes you baulk, and if further
elaboration of the set-up does not introduce any singularities, then
it seems that you should just accept the conclusions with equanimity:
Achilles is rendered immobile by conditional intentions on which no
one acts, or by a set of force fields none of which he is in direct
contact with. Bizarrely counterfactual circumstances have bizarre
consequences.



For further discussion of this case, see: Benardete (1964, which
introduces it), Moore (1990/2019), Priest (1999), Hawthorne (2000),
Perez-Laraudogoitia (2000, 2003), Yablo (2000), Oppy (2006), Koons
(2014), and Huemer (2016), Caie (2018).



We have started to see that infinity seems to be both friend and
foe--it features in powerful mathematics, but also in some vexing
conundrums. We will see more of its Manichean nature in the following
sections on probability, decision theory, and space and time. We will
also see how sophisticated methods for reclaiming it have been
developed.


## 6. Probability



Probability theory runs relatively smoothly in the finite realm, but
puzzles emerge when infinities are afoot. There are multiple sources
of infinitude, arising both in the mathematics and the interpretation
of probability. We will firstly discuss more informally these sources,
and then progress to more advanced issues.


### 6.1 Infinitude in the mathematics of probability: basics



Let us begin with the mathematics. Probability theory assumes that we
have a set of "possibilities" or "outcomes",
called a *sample space*, regarded as ways the world could be,
or the possible outcomes of a random experiment. For many purposes, an
infinite set is assumed. For example, we may toss a coin repeatedly
and be interested in how many tosses it takes until we see the first
heads. The number could be 1, or 2, or 3, or ... Here, the sample
space is denumerable. Or we might consider picking a point at random
from the [0, 1] interval of the real line--e.g., we might imagine
throwing an idealized dart at a representation of that interval, and
consider the point on which it lands. Here, the sample space is
uncountable, because it is infinitely divisible and has limits of
sequences, but bounded. Or we might consider sampling a quantity that
is governed by a normal distribution, the bell-shaped distribution
that is used to model various quantities in the real world. Here,
infinitude enters twice over: the sample space is both uncountable and
unbounded, being the entire real line.



Orthodox probability theory assigns *real numbers* between 0
and 1 (inclusive) to subsets of the sample space, and again we
encounter infinitude: there are uncountably many possible probability
values. We will soon see how we encounter it again in the way that
these values are additive.


### 6.2 Infinitude in the interpretations of probability



Infinitary considerations also enter into certain
*interpretations* of probability--attempts to explain what
probabilities are and what probability statements mean. (See the entry
on
 interpretations of probability
 for more details on what follows.) *Hypothetical frequentism*
regards probabilities as limiting relative frequencies in hypothetical
infinite sequences of trials. For example, we may toss a coin
repeatedly, generating a sequence of outcomes--e.g.



heads, tails, heads, heads, tails, tails, tails, heads, ...



We can keep track after each trial of the relative frequency of heads
so far: the ratio of the number of heads to the total number of
tosses. In our example, the sequence of relative frequencies is



\[ \frac{1}{1}, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{3}{5}, \frac{3}{6}, \frac{3}{7}, \frac{4}{8}, \ldots \]


We may then imagine this sequence extended indefinitely, and identify
the probability of heads with the limit of this sequence. However, the
very same results may be reordered, one way or another, to generate
any limiting relative frequency in [0, 1] whatsoever, if there are
infinitely many heads and infinitely many tails. Infinitude rears its
ugly head here--for a finite sequence, reordering can make no
difference to the relative frequencies of its outcomes.



According to Popper's *propensity* interpretation, a
probability \(p\) of an outcome of a certain type is a propensity of a
repeatable experiment to produce outcomes of that type with limiting
relative frequency \(p\). Again, infinitude is central to this
interpretation, and its ugly head rears as it did for hypothetical
frequentism. The *best-system* interpretation of probability,
associated with Lewis (1994) and others, is also prey to problems if
there are infinitely many events of a particular kind in the
universe--for example, infinitely many coin tosses. As Elga
(2004) shows, the interpretation's central notion of
*fit* is compromised. And even the subjective probabilities of
idealized rational agents have tacit infinitary assumptions underlying
them--for example, that the agents are logically omniscient, and
their probability assignments are infinitely sharp (single real
numbers). These assumptions have also been regarded as problematic,
especially when modeling agents who are anything like us.


### 6.3 Infinitude in the mathematics of probability: more advanced issues



To state some of the thornier puzzles generated by the mathematics of
probability, we need a more formal presentation. Kolmogorov's
(1933/1950) axiomatization begins with a finite set \(\Omega\) and an
algebra \(F\) of subsets of \(\Omega\): a set closed under
complementation and union. The members of \(\Omega\) are known as
*states* while the members of \(F\) are known as
*events*. A probability function is a function from \(F\) to
the real numbers. It is non-negative, assigns 1 to \(\Omega\), and it
is (finitely) additive--the probability that one of two mutually
exclusive events occurs is the sum of their individual
probabilities:



**Finite additivity**
  

If \(A\) and \(B\) are disjoint sets in \(F\), then \(P(A \cup B) =
P(A) + P(B)\).



Kolmogorov goes on to generalize this to an *infinite*
\(\Omega\), and to a *sigma-algebra* \(F\) of subsets of
\(\Omega\): a set closed under complementation and *countable*
union. Additivity is strengthened to hold also in infinite cases:



**Countable additivity**
  

If \(\{A\_i\}\) is a countably infinite collection of (pairwise)
disjoint sets, each in \(F\), then 
\[ P(\bigcup\_{n=1}^{\infty} A\_n) = \sum\_{n=1}^{\infty} P(A\_n) \]




Some have felt that restricting additivity to merely countable sums is
arbitrary, and is merely an artifact of the summation technique
introduced in section
 3.2.
 An alternate technique for summing infinite sets of non-negative
numbers takes advantage of the fact that a sum of nonnegative numbers
as defined earlier doesn't depend on the order of the terms. We
consider the set of all partial sums of arbitrary finite subsets of
the set, and take the least upper bound of this set to represent the
sum of the set as a whole. If this sum is some positive finite value
\(k\), then we can see that at most \(nk\) of the terms in the set
being summed can be greater than 1/\(n\). Since every positive real
number is greater than 1/\(n\) for some \(n\), this means that the set
of positive elements of the set is a countable union of finite sets,
and thus must be countable. That is, if a set being summed in this way
has uncountably many non-zero elements, the sum must be infinite.



So if we require full (unrestricted) additivity, rather than merely
countable additivity, then we can see that at most countably many
events have positive probability, and their probabilities sum to 1. A
probability distribution with these features, where events of
probability 0 have been removed, is known as a *discrete*
distribution (such as the Poisson, geometric, or negative binomial
distributions). For such a distribution, the probabilities of the
individual states determine the probabilities of all events through
the use of additivity.



However, many applications of probability require what are known as
*continuous* distributions (such as the uniform/rectangular,
normal, and beta distributions), and thus require a restriction to
countable additivity. In a continuous distribution, there are
uncountably many states, usually named by real numbers. Each
individual state has probability 0, even though events containing
uncountably many states often have non-zero probability. (This
violates full additivity.) However, in the common continuous
distributions, there is usually a way to define a probability density
for each state, such that the probability of any event is the integral
of the density over the states that make it up. In finite and discrete
distributions, it is standard to treat events of probability 0 as if
they do not happen, while in continuous distributions there is always
some event of probability 0 that occurs.



For finite and discrete distributions, there is a straightforward
definition of a concept of conditional probability. For any two events
\(A\) and \(B\), the probability of \(A\) conditional on \(B\),
notated as \(P(A\mid B)\), is defined as \(P(A \amp B)/P(B)\), if the
probability of B is non-zero, and undefined otherwise. For any fixed
\(B\), the function \(P(\\_ \mid B)\) is another probability function
on the same space. We can prove the Law of Total Probability. If
\(B\_1, B\_2, B\_3,\ldots\) form a partition (that is, every outcome is
in exactly one of the \(B\_i)\) then:

 
\[ P(A) = \sum\_i P(B\_i)P(A \mid B\_i). \]


This tells us that the unconditional probability \(P(A)\) is a
weighted average of the conditional probabilities \(P(A\mid
B\_i)\).



However, for distributions that are not discrete, so that the set of
states is uncountable in an essential way, and events of probability 0
often occur, we can't use this ratio definition of conditional
probability, since it would involve dividing by 0. However, Kolmogorov
notes (1933/1950, Ch. 5) that for any suitably nice partition, it is
still possible to come up with a definition of conditional probability
conditional on events in this partition, satisfying a generalization
of the Law of Total Probability, replacing the sum with an
integral:

 
\[ P(A) = \int P(A \mid B)dP(B) \]


(The possibility of finding a conditional probability satisfying this
integral formula is known as 'disintegrability', and it is
equivalent to a principle known as "conglomerability". For
philosophical arguments in favor of this, see Easwaran (2013b, 2019),
Rescorla (2018).) For more on determining when conditional
probabilities that satisfy this rule exist, see
Hoffmann-Jorgensen (1971), and for more on how to use
probability densities to calculate these conditional probabilities,
see Chang and Pollard (1997).



However, there are some difficulties with this account of conditional
probability. Kolmogorov notes that if the original probability space
is the uniform distribution of points on a sphere, and if \(B\) ranges
over the set of longitudes (great circles through the poles), then
probability conditional on a line of longitude will not be uniform,
but instead will be concentrated near the equator. (This fact is known
as the "Borel paradox", because Emile Borel investigated
it even before Kolmogorov's work.) Since every great circle on a
sphere can be viewed as a line of longitude with an appropriate choice
of pole, this makes the probability conditional on an event depend not
only on which event was chosen, but also which family of alternatives
it is contrasted with. (We can view each great circle as a
longitudinal line through multiple different poles, each of which
disagrees about where the equator is.)



Some have found this consequence troubling enough that they have
endorsed an alternative account of conditional probability that gives
up the Law of Total Probability, and instead insists that \(P(A|B)\)
has a unique value regardless of which alternatives to \(B\) are under
consideration. However, this also has some unpalatable consequences.
Since \(P(A)\) is no longer the average of \(P(A|B)\) where \(B\)
ranges over the elements of a partition, this means that there are
some partitions such that every element of the partition is positively
correlated with \(A\). Furthermore, the conditional probability
functions generated in this way no longer satisfy countable additivity
(Kadane, Schervish, and Seidenfeld 1996, Seidenfeld, Schervish, and
Kadane 2001, 2013).



But some, starting with de Finetti (1937, 1972, 1974) have argued on
other grounds that we should give up even countable additivity and
only accept finite additivity, with a correspondingly broader class of
probability distributions. One of de Finetti's chief arguments
involves an infinite lottery with each natural number appearing on
exactly one ticket. We would like to assign each ticket the same
probability of being drawn. Under countable additivity, this is not
possible. For if we assign probability 0 to each number's being
picked, then the sum of all these probabilities is again 0; yet the
union of all of these events has probability 1 (since it is guaranteed
that some number will be picked), and \(1 \ne 0\). On the other hand,
if we assign some (real-valued) probability \(\varepsilon \gt 0\) to
each number being picked, then the sum of these probabilities diverges
to \(\infty\), and \(1 \ne \infty\). If we drop countable additivity,
however, then we may assign 0 to each event and 1 to their union
without contradiction. In the



Supplement on God's Lottery,




we explore how an alternative approach to Kolmogorov's, a
non-Archimedean probability theory (NAP), can account for de
Finetti's lottery by assigning an infinitesimal probability to
each ticket.



However, a probability function that satisfies finite additivity
without satisfying countable additivity is mathematically much more
complicated than one that satisfies countable additivity. To even
prove the existence of such a function over the algebra of subsets of
a countable set of states requires the Axiom of Choice. With countable
additivity, it is possible to specify a discrete probability function
by enumerating the probabilities of the countably many states, and it
is possible to specify a continuous probability function by
enumerating the probabilities of the countably many rational open
sets. But if merely finite additivity is assumed, specifying a
probability function even on a countable state space may require
specifying the probabilities of uncountably many events, rather than
calculating the probabilities of these events from the countably many
probabilities of the states. Furthermore, with such probability
functions, many standard convergence results like the Strong Law of
Large Numbers fail.



For more on infinite probability spaces where only finitely additive
probability holds, see Bartha (2004), Bingham (2010), de Finetti
(1937/1989), Dubins (1975), Easwaran (2013b), Hill and Lane (1985),
Howson (2008), Kadane, Schervish, and Seidenfeld (1986), Schervish,
Seidenfeld, and Kadane (1984), Seidenfeld (2001), Seidenfeld,
Schervish, and Kadane, (2014).



A lively debate concerns a further constraint on probabilities that
may be regarded as desirable: anything that is possible should be
assigned positive probability. This is known as
*regularity*:



**Regularity**
  

If \(X\) is a non-empty subset of \(\Omega\), then \(P(X) \gt 0\).



Folk thinking about probability seems to be committed to
regularity--"if it has probability zero, it can't
happen!", as one might say.



We have seen a striking violation of regularity in de Finetti's
lottery: his assignment of 0 to *each* ticket. Regularity may
be preserved here by countably additive probabilities, but at the
expense of a uniform distribution--for example, \(\frac{1}{2}\)
to ticket 1, \(\frac{1}{4}\), to ticket 2, \(\frac{1}{8}\) to ticket
3, and so on. It may be shown that if \(F\) is uncountable, a
Kolmogorovian (real-valued) probability distribution must violate
regularity. (See e.g. Hajek 2003b.) This has led to a cottage
industry of exploring whether regularity can be preserved by allowing
the ranges of probability functions to be richer fields than the real
numbers. For example, Bernstein and Wattenberg (1969) show that there
exists a regular *hyperreal-valued* probability function for
the dart throw at [0, 1] that we imagined earlier. Each landing point
receives *infinitesimal* probability. Williamson (2007) argues
that an infinite sequence of tosses of a fair coin all landing heads
must receive probability 0 rather than some infinitesimal probability;
Howson (2019) challenges the argument. The debate for and against
preserving regularity continues, with Easwaran (2014) and Pruss (2012,
2013b, 2014) against, Benci, Horsten and Wenmackers (2012, 2016)
for--offering NAP as a way of doing so, again assigning
infinitesimal probabilities where Kolmogorov's theory would
assign 0's.



For several further puzzles involving probability in infinite spaces,
see Arntzenius, Elga, and Hawthorne (2004) and Bartha and Hitchcock
(1999). For more on infinitesimal probabilities in philosophical
applications, see Benci, Horsten and Wenmackers (2012, 2018), Easwaran
(2014), Halpern (2010), Hofweber (2014a, 2014b), Howson (2018), Kremer
(2014), Lauvers (2017), Pruss (2012, 2013, 2014, 2018a, 2018b), van
Fraassen (1976), and Wenmackers and Horsten (2013).



Infinitesimal probabilities are also appealed to in game theory. For
example, the concept of *trembling hand perfect equilibrium*
assumes that each player in a game may make a mistake with positive
but negligible probability, which may be regarded as
infinitesimal--see Halpern and Moses (2017). We will see further
use of infinitesimal probabilities in decision theory, to which we now
turn.


## 7. Decision



When you make a decision, what you choose and the way the world turns
out together determine an outcome, to which you assign a
*utility* that measures how desirable it is for you. In a
decision under certainty, each action that you may perform has exactly
one possible outcome. In that case, it seems that you should simply
perform an action that has maximal utility. (Read on, however!) In a
decision under risk, you assign probabilities to the various ways the
world could turn out--the possible *states*. Suppose that
there are various actions \(A\_i\) that you could perform, and various
states \(S\_j\) to which you assign probabilities \(p\_j\). Together
they determine outcomes to which you assign utilities \(u\_{ij}\).
Classic decision theory says that you should *maximize expected
utility*: you should perform an action that maximizes the weighted
average of the utilities associated with that action, the weights
given by your probabilities. Formally, you should maximize



\[ \text{EU}(A\_i) = \sum\_j p\_j u\_{ij} \]


(We ignore complications and variations that are irrelevant
here--see the entries on
 normative theories of rational choice: expected utility
 and
 decision theory.)



In standard cases, it is assumed that


1. there are finitely many possible actions,
2. finitely many states of the world,



and that


3. the utilities are finite.



However, we may drop each of these assumptions, yielding three
different sources of infinitude in a decision problem. Accordingly, we
will present some well-known problems that arise when one or more of
these assumptions are violated. We begin with a decision under
certainty.


### 7.1 Infinitely many possible actions: Ever-better wine



Pollock (1983) offers the following puzzle. You have a bottle of
Ever-better wine, which keeps improving as it ages: the later you open
it, the better it will be. When should you open it? There's a
good sense in which any time is too soon: opening it slightly later
would be better. But the worst option is never to open it, and to
avoid this it must be opened at some time. This decision problem has
uncountably many possible actions, but we could easily make them
denumerable by adding that the bottle can only be opened at discrete
times--e.g. on the hour. You would gladly perform an action that
has maximal utility, but here there is no such action! This problem
displays an intriguing feature that Bartha, Barker and Hajek
(2013) call *discontinuity at infinity*: "an infinite
sequence of choices, each apparently sanctioned by plausible
principles, converges ... to a 'limit choice' whose
utility is distinct from, and typically much lower than, the limit
approached by the utilities of the choices in the sequence"
(630). Their paper discusses other decision problems with this
feature. For more discussion of this kind of phenomenon, see Chow,
Robbins, and Siegmund (1971) and Seidenfeld (1981).


### 7.2 Infinitely many states: the St. Petersburg paradox



A fair coin is tossed. If it comes up heads, you receive $2. If it
comes up tails, the coin is tossed for a second time. If it then comes
up heads, you receive $4. If it comes up tails, the coin is tossed for
a third time. If it then comes up heads, you receive $8. If it comes
up tails, the coin is tossed for a fourth time. And so on. We continue
until the coin comes up heads. If this takes n tosses, then you win
$\(2^n\).



How much should you be prepared to pay to play this game? You have a
1/2 chance of winning $2; and a 1/4 chance of winning $4; and a 1/8
chance of winning $8; ....; and a \(1/2^n\) chance of winning
$\(2^n\); and ... . Hence, your expected payoff from playing the
St. Petersburg game is infinite:

 
\[\begin{align}
(\frac{1}{2}\times 2) &+ (\frac{1}{4} \times 4) + (\frac{1}{8} \times 8) + \cdots + (\frac{1}{2^n} \times 2^n) + \cdots \\
 &= 1 + 1 + 1 + \cdots 
\end{align}\]


If we identify dollars won with utilities, the game has infinite
expected utility.



Decision theory seems to say that you should be prepared to pay any
finite amount to play this game. But most people think this is crazy;
indeed, most would only pay a few dollars to play (Neugebauer 2010).
And decision theory seems to say that you should be prepared to pay
any finite amount for a ticket in any finite lottery whose payoff is a
single play of this game. That may seem really crazy.



You might object that the utility of money decreases as you obtain
more of it: if the rate of this decrease is sufficiently large, then
the expected value of playing the game is finite. Daniel Bernoulli
argued that utility goes by the logarithm of the amount of money, and
indeed replacing the dollar amounts with their logarithms yields a
finite expected utility. However, we can retell the story in terms of
utilities themselves. And we can retell it with super-exponential
escalation of the value of the payoffs: taking logarithms then gives
us exactly the original expected utility. (See Menger 1967/1934.) In
fact, as long as utilities are unbounded, we can fashion a version of
the game that has infinite expected utility.



So you might object that the utilities are bounded. (See e.g., Hardin
1982.) However, unbounded quantities abound--length, volume,
mass, curvature, temperature, and so on. Why is utility unlike them in
this regard? Moreover, one might imagine a case in which utilities are
intimately linked to another such quantity--e.g., the further
away you get from some undesirable place, the better--and an
unbounded function might link them. Moreover, as we have noted,
probability theory is already shot through with infinitude; we need a
principled reason why *this* kind of infinitude should be
shunned. (See Nover and Hajek 2004 for further discussion.) And
perhaps it is not crazy after all to value the St. Petersburg game
infinitely. After all, it dominates each truncation of the game, which
pays nothing if heads has not come up after \(n\) trials (for each
\(n\)): the St. Petersburg game's outcome is equally good in
finitely many states, and strictly better in infinitely many.
Plausibly, then, it should be preferred to all these truncations of
the game (Hajek and Nover 2006, 2008)--its desirability is
greater than \(n\), for each \(n\).



For further discussion of the St. Petersburg game, see, for example:
Samuelson (1977), Jeffrey (1983), Weirich (1984), Cowen and High
(1988), Jordan (1994), Chalmers (2002), Peters (2011) and the entry on
 the St. Petersburg paradox.



Related but different problems arise in the *Pasadena game*, a
St. Petersburg-like game in which the expected payoff is apparently
*undefined* (rather than infinite). Then, it seems that
decision theory goes silent regarding the value of the game. And yet
various choices regarding the game seem to be rationally
required--e.g. preferring the game plus a $1 to the game itself.
For further discussion, see e.g. Nover and Hajek (2004),
Hajek and Nover (2006, 2008), Hajek (2014), Easwaran
(2008), Bartha (2016) and Colyvan and Hajek (2016).


### 7.3 Infinite utility: Pascal's Wager



In the St. Petersburg game, each possible payoff is finite; it is the
way in which they are averaged by the expected utility formula that
yields the infinitude. We now turn to a classic decision problem in
which a possible payoff itself is infinite.



Pascal maintains that we cannot know whether God exists or not, but he
argues that we can solve the decision problem of whether or not to
'wager for God'--roughly, to cultivate belief in God.
There are two available courses of action: wager for God, or fail to
wager for God. There are two relevant conceivable states of the world:
God exists, or God does not exist. The probability that God exists is
\(p\), whence the probability assigned that God does not exist is \(1
- p\). The utility of wagering for God, if God exists--salvation
forever--is infinite. All of the other utilities--of an
earthly life of some finite duration--are finite. We may
formulate the resulting decision table as follows:




|  |  |  |
| --- | --- | --- |
|  | *God exists* | *God does not exist* |
| Probabilities: | \(p\) | \(1-p\) |
| *Wager for God* | \(\infty\) | \(f\_1\) |
| *Wager against God* | \(f\_2\) | \(f\_3\) |



We may now do the expected utility calculations:



The expected utility of wagering for God is

 
\[ p\cdot \infty + (1 - p)\cdot f\_1 = \infty. \]


The expected utility of failing to wager for God is

 
\[ p \cdot f\_2 + (1 - p)\cdot f\_3 = \text{ a finite value}. \]


In order to maximize expected utility, one ought to wager for God.



Among the many objections that have been levelled at Pascal's
Wager, several focus on the role that '\(\infty\)' plays
in the argument. Can utilities be infinite? There is a considerable
literature that considers possible extensions of our decision rule,
and possible modifications to the framework within which the decision
problem is framed. However, to date, there is no widely accepted
alternative formulation of Pascal's Wager that avoids all these
difficulties that focus on the role that '\(\infty\)'
plays in the argument. And once infinite utilities are countenanced,
it seems that we should be open to infinitesimal probabilities also.
But then there is the prospect that when an infinite utility and an
infinitesimal probability are multiplied in the expected utility
formula, the product may be a finite number. Will wagering for God
still maximize expected utility? These issues and more are discussed
in the entry on
 Pascal's wager.



For further discussions of the treatment of infinity in Pascal's
Wager, see, for example: Duff (1986), Oppy (1991, 2018), Hajek
(2003a, 2018), Bartha (2007, 2018), Bartha and Pasternack (2018),
Monton (2011), and Wenmackers (2018).


### 7.4 Infinite utility streams



So far we have been considering decisions in which one's payoff
comes in a single 'hit': a reward (or punishment) comes
all at once. However, we can also consider cases in which one is to
choose between different infinite utility *streams*--e.g.,
streams of finite daily utility that accumulate over an infinite
future. There is an obvious candidate for evaluating the utility of a
finite stream: add the utilities along the stream. But this method is
not available when we have an infinite stream; we require additional
principles to help us evaluate such streams, and it is not obvious
what those principles should be.



Suppose that a day spent in Heaven has utility 1 and a day spent in
Hell has utility \(-1\). Suppose further that, for all \(n\), \(n\)
days in Heaven have utility \(n\), and \(n\) days in Hell have utility
\(-n\). Suppose, finally, that, for all \(m\) and \(n\), any
combination of \(m\) days in Heaven and \(n\) days in Hell has utility
\(m - n\).



Here are some candidate principles for the comparison of alternative
possible future utility streams:


1. One should prefer the possible future utility stream that has
maximal total utility (if there is one).
2. If more than one possible future utility stream has divergent
utility--i.e. is such that there is no finite value to which the
total utility of the stream converges as the number of days
increases--one should prefer the divergent utility stream whose
partial sum is *dominant* (if there is one). This means that on
some days, the sum of utility to that day is greater than for any
other stream, and on no days, the sum of utility to that day is less
than for some other stream.
3. If more than one possible future utility stream has divergent
utility, one should be indifferent between divergent future utility
streams that are permutations of one another.



Consider the choice between the following two infinite utility
streams:


1. An infinite number of days in Heaven.
2. An infinite number of days in Heaven preceded by a finite number
of days in Hell.



Principle 2 says correctly that we should prefer (a) to (b).



However, consider the choice between the following two options:


3. An infinite number of alternating days, first in Heaven, and then
in Hell.
4. An infinite number of alternating days, first in Hell, and then in
Heaven.



Principle 2 says, incorrectly presumably, that we should prefer (c) to
(d).



Now consider the choice between the following two options:


5. An infinite number of alternating days, first in Heaven, and then
in Hell.
6. An infinite number of alternating days, first one in Heaven and
one in Hell, then two in Heaven and one in Hell, then three in Heaven
and one in Hell, and so on.



While Principle 3 says, (perhaps) correctly that we should be
indifferent between (c) and (d), it also says, (surely) incorrectly,
that we should be indifferent between (e) and (f).



In the face of these difficulties, you might consider weakening the
principles:


4. If more than one possible future utility stream has divergent
utility, prefer the divergent possible future utility stream that is
step-by-step dominant (if there is one).
5. If more than one possible future utility stream has divergent
utility, maintain indifference between divergent possible future
utility streams that are finite permutations of one another (i.e. that
can be derived from one another by finitely many exchanges at
neighbouring steps).



But this pair of principles yields no verdict in the case of (e) and
(f), and so does not yield a complete set of principles.



More generally, it is hard to codify rules for choosing among infinite
utility streams. Indeed, there are some impossibility results in the
economics literature which suggest that there is no fully satisfactory
theory that countenances them.



For further discussion of infinite utility streams, see, for example:
Segerberg (1976), Jeffrey (1983), Nelson (1991), Vallentyne (1993,
1994, 1995), Cain (1995), Ng (1995), Van Liedekerke (1995), Lauwers
(1997a, 1997b, 1997c, 1997d), Vallentyne and Kagan (1997), Basu and
Mitra (2003), Crespo, Nunez, and Rincou-Zapatero (2009),
Bartha, Barker and Hajek (2014), and Jonsson and Voorneveld
(2015).



Each of these decisions problems wears its infinitude on its sleeve:
it is obvious that there are infinitely many possible actions, or
infinitely many states, or infinite utility, or infinite streams of
utility. However, in some problems, such infinitude is not
foregrounded, but it is tacitly there nonetheless. The
*two-envelope paradox* is such a problem. See the



Supplement on Tacitly Infinite Decision Problems: Two Envelopes.



There are various other paradoxes of infinity in decision
theory--the interested reader may follow these references:


* "An Infinite Decision Puzzle": Barrett and Arntzenius
(1999)
* "Trumped": Arntzenius and McCarthy (1997)
* "Rouble Trouble": Arntzenius and Barrett (1999)
* "The Airtight Dutch Book": McGee (1999)
* Several paradoxes in Arntzenius, Elga and Hawthorne (2004)
* "The Cable Guy": Hajek (2005).


## 8. Space and time



Considering whether space and time are infinite in extent and
divisibility has led to many famous puzzles, paradoxes and antimonies.
It was on account of such paradoxes that Kant was led to the claim
that whether space is finite or infinite escapes any possible
empirical determination. Kant's presentation of the antinomies
rested on a number of assumptions (such as the distinction between
infinite and unbounded) that were undermined by later results in
mathematics or were simply found to be philosophically questionable.
Another interesting paradox relates to divisibility. In this section
we discuss Kant on the antinomies of space and time and a
measure-theoretic solution to this paradox of divisibility. This is
followed by a quick overview of some developments in non-Euclidean
geometries and relativistic cosmology. In the final part, we mention
some recent developments in cosmic topology, an area of cosmology that
attempts to determine whether space is finite or infinite by a
combination of empirical observation and mathematical theorizing. The
emphasis will be on the latter aspect.


### 8.1 Antinomies of space and time



Many philosophers have devised paradoxes and even putative
'antinomies' that exploit the structural features of space
and time in a way that essentially involves the infinite. Among the
ancients, Zeno is renowned for his paradoxes of space, time and
motion. They involve infinitely many spatial or temporal subdivisions
or processes that are putatively impossible--see the entry on
 Zeno's paradoxes.
 Among the moderns, Kant is particularly notable for his treatment of
the extent of space and time in his 'First Antinomy of Pure
Reason'. We turn to it now.


#### 8.1.1 Kant



In the *Critique of Pure Reason*, A426-A434,
B454-B462--Kant gives 'proofs' of conflicting
theses--'thesis' and
'antithesis'--about the extent of space and time. The
'thesis' says that:


1. the world has a beginning in time; and
2. the world has a finite extension.



The 'antithesis' says:


3. the world has no beginning in time; and
4. the world has infinite extension.



To a reasonable approximation, the 'proofs' run as
follows:


1. If the world has no beginning in time, then, up to any given
moment, an eternity has elapsed: there has passed away an infinite
series of successive states of things. But the infinity of a series
consists in the fact that it can never be completed through successive
synthesis. So it is impossible for an infinite series of successive
states of things to have passed away: the world has a beginning in
time.
2. Since infinite extension cannot be thought in a single completed
act of thought, the world can only be thought to have infinite
extension through an act of synthesis in which completion is achieved
via the addition of units. But an act of synthesis that achieved
completion via the addition of units requires the lapse of an infinite
amount of time--and we have already seen, in (a), that this is
impossible. Consequently, it cannot be thought that the world has
infinite extension. So the world does not have infinite
extension.
3. Something begins to exist only if there is a prior time at which
it does not exist. Hence, if the world has a beginning in time, there
must be an earlier time at which it does not exist: an empty time. But
nothing can come into existence in an empty time, since there is no
sufficient reason for the thing to come into existence in one rather
than another part of empty time. So the world does not have a
beginning in time.
4. If the world has finite extension, then the world is contained in
an unlimited empty space. Consequently, the objects in the world are
not only related *in* space but also related *to* space.
In particular, the relation of the world to empty space is a relation
of the world to no object, i.e. to nothing. But there can be no such
relation. So the world has infinite extension.



Much in Kant's discussion of the antinomies of space and time is
marred by his conflation of the modern definition of infinity as lack
of finitude with the Aristotelian conception of the impossibility of
completion. In addition, at A487/B515 we have confirmation that Kant
uses "infinite" and "unbounded", as well as
"finite" and "bounded", synonymously:
"For if it [the magnitude of the world in space] is infinite and
unbounded, then it is too big for every possible empirical concept. If
it is finite and bounded, then you can rightfully ask: What determines
this boundary?" It was only with the work of Bernhard Riemann in
the nineteenth century that geometrical concepts of space were
introduced that allowed the decoupling of unboundedness and infinity
(and correspondingly of bounded and finite). See section
 8.2.



You can find further--sometimes sympathetic--discussion of
these arguments in Bennett (1966), Huby (1971), Whitrow (1978), Craig
(1979), Moore (1990/2019), Oppy (2006), Huemer (2016), and the entry
on
 Kant's Critique of Metaphysics.


#### 8.1.2 Measure



Here is a Zeno-style argument:



Suppose for *reductio* that a finite line segment of non-zero
length is composed of infinitely many disjoint parts of equal,
real-valued length.


1. Either the parts all have zero length or they all have the same
non-zero length.
2. The length of the whole line segment is the sum of the lengths of
the parts.
3. If the parts all have zero length, then the line segment has zero
length, contradicting our assumption that it has non-zero length.
4. If the parts all have non-zero length, then the line segment has
infinite length contradicting our assumption that it is finite.



Conclusion 1: A finite line segment cannot be composed of infinitely
many disjoint parts of equal real-valued length.



Therefore,



Conclusion 2: It cannot be composed of points.



Premise 1 is beyond reproach. However, premises 2, 3, and 4 require us
to be careful about how lengths add. Recall that in section
 3.2
 we discussed how to add a countable sequence of numbers - but
the method described there depends on the order, and requires a
countable, well-ordered sequence. Although there are techniques for
summing uncountable sets of non-negative numbers, most mathematicians
deny that lengths or other measures can be added in these ways. This
is parallel to what Kolmogorov said about probability (see section
 6.3).
 Probability and length are two paradigms of the more general
mathematical field of "measure theory", which includes all
such countably additive real-valued functions. For a more detailed
discussion of this problem, including approaches involving
infinitesimal length, see Skyrms (1983).



For more about measure theory, see Bartle (1995) and Tao (2011).


### 8.2. Non-Euclidean geometries, relativistic space-time, and cosmic topology


#### 8.2.1 Non-Euclidean geometries



In section
 1
 we anticipated that Archytas' argument for the infinitude of
the cosmos, and Kant's treatment of the antinomies, conflated
the notions of *finiteness* and *boundedness*.



We now need to introduce another aspect of 19th century mathematics
that brought that crucial distinction into focus. The distinction
between finiteness and boundedness (and consequently that between
infinitude and unboundedness) greatly improved our understanding of
issues concerning the structure of space, and what shape a finite, or
an infinite space, might take. Recall that for two centuries after
Newton, cosmology was developed within the framework of Euclidean
infinite space. Such a space is infinite in all directions, it is
homogeneous and isotropic--that is, it is the same at all
locations and in all directions.



In the middle of the 19th century, alternative conceptions of
geometrical space were developed, so-called *non-Euclidean
geometries*. Gauss, Bolyai, Lobachevski and Riemann, showed that
one can develop geometries that falsify Euclid's parallel axiom
while preserving all the other Euclidean axioms. The axiom states (in
a later but equivalent version to the one given by Euclid) that for
any line and a point external to that line, there is one and only one
parallel to the given line passing through the point. The statement
contains a claim of existence and a claim of uniqueness. It can be
thus falsified by denying existence or by accepting the existence but
denying the uniqueness. Both alternatives have been developed, with
some of the earliest interpretations using surfaces. The first
alternative, where no parallels exist to any given line that pass
through a given point external to the given line, is known as
*elliptic geometry*. An instance of elliptic geometry is
*spherical geometry*, so called because it can be modeled on
the surface of a sphere. The second alternative is known as
*hyperbolic* geometry and in it for every line in the model and
any point outside of the line there are infinitely many parallels to
that line passing through the point. A portion of the surface of a
horse saddle can be used to model hyperbolic geometry. (The pictures
below are based on those in Luminet 2008: 49.)



![Illustration showing one of three different surfaces (a sphere, a plane, and a saddle) representing the notion of curvature of a surface at a point p of the surface. The illustration displays how the three surfaces can be used to to model various geometries (elliptic, euclidean, and hyperbolic) each of which is characterized by a different curvature of the surface, positive for elliptic, null for Euclidean, and negative for hyperbolic geometry. Here, a triangle is shown on a sphere surface and the sum of the interior angles of the triangle is larger than 180 degrees.](spherical.png)
  

![Here, a triangle is shown on a plane and the sum of the interior angles of the triangle is 180 degrees.](euclidean.png)
  

![Here, a triangle is shown on a saddle surface and the sum of the interior angles of the triangle is less than 180 degrees.](hyperbolic.png)


The curvature of a surface at a point \(p\) on the surface measures
how much the surface bends away from its tangent plane at \(p\). The
curvature of a surface is constant if at every point \(p\) of the
surface the surface bends away from the tangent plane by the same
quantity. Examples of surfaces which can be used to model various
geometries are the surface of the cylinder (Euclidean; constant
curvature 0), the sphere (spherical; positive curvature), and the
horse saddle (hyperbolic; negative curvature). They are all homogenous
and isotropic but they have different constant curvature.



Such geometries on surfaces spurred the development of
three-dimensional and higher dimensional spaces with different
curvatures: positive, null, and negative. An example of a space of
positive constant curvature is the *3-sphere* (also called
hypersphere) used by Bernhard Riemann in his 1854 dissertation (see
Riemann 1868; for an English translation see Riemann 2016). A
*3-sphere* is a surface in a four-dimensional space that is
obtained as a generalization of the 2-*sphere* as visualized in
three dimensions: in both cases we define the relevant notion as a
locus of points that have a constant distance from a point (its
center). For instance, the unit 2-sphere centered at the origin is the
set of triples of real numbers \((x, y, z)\) that are one unit away
from (0, 0, 0), i.e. that satisfy \(x^2 +y^2 +z^2 =1\), and the
3-sphere with distance 1 from the origin (0, 0, 0, 0) is the set of
quadruples \((x, y, z, w)\) of real numbers satisfying \(x^2 +y^2 +z^2
+w^2 =1\). It is a model of physical space that is *finite but
unbounded*, in explicit opposition to Newton's conception of
space.



Archytas's argument (in section
 1
 above), which conflated unboundedness with infinitude, could finally
be put to rest. Riemann's model allows for the universe to be
finite and unbounded at the same time. In 1854 he wrote: "The
unboundedness of space possesses in this way a greater empirical
certainty than any external experience. But its infinite extent by no
means follows from this; on the other hand if we assume independence
of bodies from position, and therefore ascribe to space constant
curvature, it must necessarily be finite provided this curvature has
ever so small a positive value. If we prolong all the geodesics
starting in a given surface-element, we should obtain an unbounded
surface of constant curvature, i.e., a surface which in a flat
manifoldness of three dimensions would take the form of a sphere, and
consequently be finite." (Riemann 2016: 39)



The distinction between infinite and unbounded is an integral part of
the conceptual leap that leads to the idea that physical space need
not be Euclidean. In the following section we will briefly describe
how issues of curvature and topology play a role in addressing the
question of whether the world is spatially finite or infinite in
cosmology.



On non-Euclidean geometries the reader will find useful Greenberg
(2007) and Gray (2010). On the philosophical relevance of curvature
and Riemannian geometry see the classic Torretti (1984).


#### 8.2.2 Relativistic space-time and cosmic topology



In 1915 Einstein introduced general relativity, and our conception of
the universe is based on it. General relativity rests on a conception
of space and time--or better, space-time matter--that stands
in opposition to the Newtonian one we have described above. In
Einstein's theory, space-time is deformable and its shape
depends on the presence of matter. Space-time is, in technical terms,
a four-dimensional manifold. We may think of an \(n\)-dimensional
manifold as a set of \(n\)-tuples of real numbers. The spatial section
of a four-dimensional manifold of space-time is a three-dimensional
manifold (one can think of it as a set of triples of real numbers),
and when cosmologists ask about the shape of the universe they try to
characterize this three-dimensional manifold. The curvature of
space-time corresponds to gravitation, and light rays and other
material particles follow the geodesics (shortest paths) in the
manifold. In general, the geodesics differ depending on the
matter-energy content of the space being considered. The geodesics of
the surface of the sphere (i.e. a two-dimensional surface) are
portions of great circles. In the Euclidean plane, they are segments
of straight lines. There are analogous notions for four-dimensional
manifolds. Einstein's equations for general relativity describe
how the matter-energy content of the universe determines the geometry
of space-time. The equations also yield cosmological models, which
must be tested by empirical observation. The equations allow for
multiple solutions and, as Alexander Friedmann (1924) observed,
"in the absence of additional hypotheses, Einstein's
equations for the universe do not allow to definitely answer the
question of the finiteness of the universe". Let us briefly
explain what is at stake in this comment, by pointing out the role of
curvature and topology with respect to the issue of finiteness vs.
infiniteness of our universe in relativistic cosmology. Topology is
the branch of geometry that classifies spaces according to whether
they can be transformed into one another "continuously",
that is with transformations that do not lead to cuts or tears.



In 1917, Einstein posited a static finite universe. The finiteness was
given by the choice of the 3-sphere (see section
 8.2.1)
 and the static nature of the universe by the fact that the radius of
the hypersphere did not change with time. With Friedmann
(1922-24) and Lemaitre (1927), Einstein's static
model would be replaced by dynamical models (to account for the
empirical evidence that by 1930 led to the postulation that the
universe is expanding, i.e., most galaxies, galaxy clusters, etc. are
growing further apart, just as spots on an uninflated balloon grow
further apart when the balloon is blown up). Such models are also
among the possible solutions for Einstein's equations and they
were the source for the so-called "Big Bang" theories. But
what about finiteness? The finiteness or infiniteness of the universe
are not determined by Einstein's equations, which allow for both
possibilities. In his choice of the 3-sphere, Einstein was motivated
by considerations that had to do with preserving a hypothesis by Mach
on inertial mass and inertial motion. Friedmann and Lemaitre
also opted for the finiteness of the universe (we need not get into
why they did so). Their dynamical models assumed a uniform
distribution of matter in the universe and that space is homogeneous
and isotropic. But the Friedmann-Lemaitre dynamical solutions
still allow for a great variety of mathematical solutions and do not
settle the issue of finiteness. Our observations in what follows are
restricted to such models.



Space, in this context, is characterized by its curvature (taken to be
constant) and its topology. Let us consider curvature first. In these
models, there are three possible types of spaces depending on whether
the curvature of the space is negative, null, or positive. The spaces
corresponding to such curvatures are called *hyperbolic*,
*Euclidean*, and *elliptical*. A spherical space
(constant positive curvature) is always of finite extension, no matter
what its topology is. This explains, at least in part, why many early
cosmologists (including Einstein, de Sitter, Friedmann, Lemaitre
and others) opted for this solution. Indeed, for a long time issues
concerning the topology of space were not brought to the fore due to
the implicit assumption that the topology of space was a simply
connected topology (in a simply connected topology every loop on the
surface can be continuously contracted to a point). Under that
assumption, spaces of positive constant curvature are finite and those
of null and negative constant curvature are infinite. The issue then
of the finiteness vs. infiniteness of the universe rests on the mean
density of matter and energy and on the value of a parameter
\(\lambda\) introduced by Einstein in 1917, called the
*cosmological constant* (measuring a sort of anti-gravitational
force). With most cosmologists (but not Einstein in 1917) assuming
\(\lambda = 0\), and with the assumption that space is simply
connected, determining curvature (and hence resolving the finiteness
vs infinity issue) depends only on a critical value for the mean
density of matter--equivalently, on a density parameter
\(\Omega\). Thus, under those assumptions, it would be in principle
possible to determine the curvature of space experimentally.



Different values of \(\lambda\) lead to different scenarios for the
evolution of the universe. With \(\lambda = 0\), if the curvature of
space is negative or null we end up with a constantly expanding
universe; if the curvature of space is positive, a phase of expansion
would be followed by a contraction leading to a "big
crunch". Other values of the cosmological constant are possible
and if \(\lambda \lt 0\) a "big crunch" will occur no
matter what the curvature of space is. By contrast, if \(\lambda \gt
0\), no "big crunch" will occur. New experimental evidence
(coming from type 1A supernovae and fossil radiation) seems to
indicate a positive mean density of matter and a value of \(\lambda
\gt 0\). In this case the universe would be finite while still
remaining in perpetual accelerated expansion.



Moreover, recent work has pointed out the importance of considering
*multiply connected* topologies. Unlike what happens for simply
connected topologies, curvature does not immediately determine the
finiteness or infinity of the space. Indeed, there are spaces of null
or negative curvature that can be finite or infinite depending on the
multiply connected topology associated to them. This leads into
*cosmic topology*, which investigates the global shape of space
and how it can be determined experimentally. If space has positive
curvature, then the universe is finite independently of the specific
topology associated with it. But if the curvature is negative or null,
whether the universe is finite or not will depend on the topology.
Thus, determining whether the universe is finite or infinite requires
not only determining the mean density of matter (which determines the
curvature of space) but also the topology of space. Two major
techniques that are employed experimentally to determine the topology
of space are cosmic crystallography and the circles in the sky method
(based on the cosmic microwave background).



For further information on cosmic topology see Luminet, Starkmann, and
Weeks, (1999), Luminet and Lachieze-Rey (2005), Luminet (2005)
(English 2008). See also Aguirre (2011) and Luminet (2015). For more
technical treatment see Thurston (1997), Weeks (2001), and Hitchmann
2018.


## 9. Conclusion



We are well aware that our discussion of infinity is
incomplete--but then, so is any such discussion. We take some
comfort from the fact that it is impossible to give balanced coverage
to a boundless set of issues in finite space.



There are many more philosophically significant paradoxes and puzzles
that involve infinity in one way or another; we have only given a
small sample. And new paradoxes involving infinity seem to appear at
an ever-increasing rate (doubtless yet another one can be fashioned
out of this very fact!). However, so too are our tools for
understanding infinity. Of course, we cannot give a definitive
assessment of the state of play, but the theoretical developments that
we have sketched and references that we have cited make us sanguine
that overall, the prospects for our relationship with infinity are
good: we can indeed live with it.