## 1. Emergence



The concept of a set appears deceivingly simple, at least to the
trained mathematician, and to such an extent that it becomes difficult
to judge and appreciate correctly the contributions of the pioneers.
What cost them much effort to produce, and took the mathematical
community considerable time to accept, may seem to us rather
self-explanatory or even trivial. Three historical misconceptions that
are widespread in the literature should be noted at the outset: 


1. It is not the case that actual infinity was universally rejected
before Cantor.
2. Set-theoretic views did not arise exclusively from analysis, but
emerged also in algebra, number theory, and geometry.
3. In fact, the rise of set-theoretic mathematics preceded
Cantor's crucial contributions.



All of these points shall become clear in what follows.



The notion of a collection is as old as counting, and logical ideas
about classes have existed since at least the "tree of
Porphyry" (3rd century CE). Thus it becomes difficult
to sort out the origins of the concept of set. But sets are neither
collections in the everyday sense of this word, nor
"classes" in the sense of logicians before the
mid-19th century. The key missing element is objecthood
-- a set is a mathematical object, to be operated upon just like
any other object (the set \(\mathbf{N}\) is as much 'a
thing' as number 3). To clarify this point, Russell employed the
useful distinction between a class-as-many (this is the traditional
idea) and a class-as-one (or set). 



Ernst Zermelo, a crucial figure in our
story, said that the theory had historically been "created by
Cantor and Dedekind" [Zermelo 1908, 262]. This suggests a good
pragmatic criterion: one should start from authors who have
significantly influenced the conceptions of Cantor, Dedekind, and
Zermelo. For the most part, this is the criterion adopted here.
Nevertheless, as every rule calls for an exception, the case of
Bolzano is important and instructive, even though Bolzano did not
significantly influence later writers.



In 19th century German-speaking areas, there were some
intellectual tendencies that promoted the acceptance of the actual
infinite (e.g., a revival of Leibniz's thought). In spite of
Gauss's warning that the infinite can only be a manner of
speaking, some minor figures and three major ones (Bolzano, Riemann,
Dedekind) preceded Cantor in fully accepting the actual infinite in
mathematics. Those three authors were active in promoting a
set-theoretic formulation of mathematical ideas, with Dedekind's
contribution in a good number of classic writings (1871, 1872,
1876/77, 1888) being of central importance.



Chronologically, Bernard Bolzano was the first, but he exerted almost
no influence. The high quality of his work in logic and the
foundations of mathematics is well known. A book entitled
*Paradoxien des Unendlichen* was posthumously published in
1851. Here Bolzano argued in detail that a host of paradoxes
surrounding infinity are logically harmless, and mounted a forceful
defence of actual infinity. He proposed an interesting argument
attempting to prove the existence of infinite sets, which bears
comparison with Dedekind's later argument (1888). Although he
employed complicated distinctions of different kinds of sets or
classes, Bolzano recognized clearly the possibility of putting two
infinite sets in one-to-one correspondence, as one can easily do,
e.g., with the intervals \([0, 5]\) and \([0, 12]\) by the function
\(5y = 12x\). However, Bolzano resisted the conclusion that both sets
are "equal with respect to the multiplicity of their
parts" [1851, 30-31]. In all likelihood, traditional ideas
of measurement were still too powerful in his way of thinking, and
thus he missed the discovery of the concept of cardinality (however,
one may consider Non-Cantorian ideas, on which see Mancosu 2009). 



The case of Bolzano suggests that a liberation from metric concepts
(which came with the development of theories of projective geometry
and especially of topology) was to have a crucial role in making
possible the abstract viewpoint of set theory. Bernhard Riemann
proposed visionary ideas about topology, and about basing all of
mathematics on the notion of set or "manifold" in the
sense of class (*Mannigfaltigkeit*), in his celebrated
inaugural lecture "On the Hypotheses which lie at the
Foundations of Geometry" (1854/1868a). Also characteristic of
Riemann was a great emphasis on *conceptual* mathematics,
particularly visible in his approach to complex analysis (which again
went deep into topology). To give but the simplest example, Riemann
was an enthusiastic follower of Dirichlet's idea that a function
has to be conceived as an arbitrary correspondence between
numerical values, be it representable by a formula or not; this meant
leaving behind the times when a function was defined to be an analytic
expression. Through this new style of mathematics, and through his
vision of a new role for sets and a full program for developing
topology, Riemann was a crucial influence on both Dedekind and
Cantor (see Ferreiros 1999).



The five-year period 1868-1872 saw a mushrooming of
set-theoretic proposals in Germany, so much so that we could regard it
as the birth of set-theoretic mathematics. Riemann's geometry
lecture, delivered in 1854, was published by Dedekind in 1868, jointly
with Riemann's paper on trigonometric series (1854/1868b, which
presented the Riemann integral). The latter was the starting point for
deep work in real analysis, commencing the study of
"seriously" discontinuous functions. The young Georg
Cantor entered into this area, which led him to the study of
point-sets. In 1872 Cantor introduced an operation upon point sets
(see below) and soon he was ruminating about the possibility to
iterate that operation to infinity and beyond: it was the first
glimpse of the transfinite realm.



Meanwhile, another major development had been put forward by Richard
Dedekind in 1871. In the context of his work on algebraic number
theory, Dedekind introduced an essentially set-theoretic viewpoint,
defining fields and ideals of algebraic numbers. These ideas were
presented in a very mature form, making use of set operations and of
structure-preserving mappings (see a relevant passage in
Ferreiros 1999: 92-93; Cantor employed Dedekind's
terminology for the operations in his own work on set theory around
1880 [1999: 204]). Considering the ring of integers in a given field
of algebraic numbers, Dedekind defined certain subsets called
"ideals" and operated on these sets as new objects. This
procedure was the key to his general approach to the topic. In other
works, he dealt very clearly and precisely with equivalence relations,
partition sets, homomorphisms, and automorphisms (on the history of
equivalence relations, see Asghari 2018). Thus, many of the usual
set-theoretic procedures of twentieth-century mathematics go back to
his work. Several years later (in 1888), Dedekind would publish a
presentation of the basic elements of set theory, making only a bit
more explicit the operations on sets and mappings he had been using
since 1871.



The following year, Dedekind published a paper [1872] in which he
provided an axiomatic analysis of the structure of the set
\(\mathbf{R}\) of real numbers. He defined it as an ordered field
that is also complete (in the sense that all Dedekind-cuts on
\(\mathbf{R}\) correspond to an element in \(\mathbf{R}\));
completeness in that sense has the Archimedean axiom as a consequence.
Cantor too provided a definition of \(\mathbf{R}\) in 1872,
employing Cauchy sequences of rational numbers, which was an elegant
simplification of the definition offered by Carl Weierstrass in his
lectures. The form of completeness axiom that Weierstrass preferred
was Bolzano's principle that a sequence of nested closed
intervals in \(\mathbf{R}\) (a sequence such that
\([a\_{m+1},b\_{m+1}] \subset [a\_{m},b\_{m}]\)) "contains" at
least one real number (or, as we would say, has a non-empty
intersection).



The Cantor and Dedekind definitions of the real numbers relied
implicitly on set theory, and can be seen in retrospect to involve the
assumption of a Power Set principle. Both took as given the set of
rational numbers, and for the definition of \(\mathbf{R}\) they relied
on a certain totality of infinite sets of rational numbers (either the
totality of Cauchy sequences, or of all Dedekind cuts). With this,
too, constructivistic criticism of set theory began to emerge, as
Leopold Kronecker started to make objections to such infinitary
procedures. Simultaneously, there began a study of the topology of
\(\mathbf{R}\), in particular in the work of Weierstrass, Dedekind,
and Cantor. The set-theoretic approach was also exploited by several
authors in the fields of real analysis and complex analysis (e.g.,
Hankel, du Bois-Reymond, H.J.S. Smith, U. Dini) and by Dedekind in
joint work with Weber (1882), pioneering algebraic geometry.



Cantor's derived sets are of particular interest (for the
context of this idea in real analysis, see e.g., Dauben 1979, Hallett
1984, Lavine 1994, Kanamori 1996, Ferreiros 1999). Cantor took
as given the "conceptual sphere" of the real numbers, and
he considered arbitrary subsets \(P\), which he called
'point sets'. A real number \(r\) is called a
*limit point* of \(P\), when all neighbourhoods of \(r\)
contain points of \(P\). This can only happen if \(P\) is
infinite. With that concept, due to Weierstrass, Cantor went on to
define the *derived set* \(P'\) of \(P\), as the *set of
all* the limit points of \(P\). In general \(P'\) may be infinite
and have its own limit points (see Cantor's paper in Ewald
[1996, vol. 2, 840ff], esp. p. 848). Thus one can iterate the
operation and obtain further derived sets \(P''\), \(P'''\)...
\(P^{(n)}\) ... It is easy to give examples of a set \(P\) that
will give rise to non-empty derived sets \(P^{(n)}\) for all finite
\(n\). (A rather trivial example is \(P = \mathbf{Q}\_{[0,1]}\), the
set of rational numbers in the unit interval; in this case \(P' =
[0,1] = P''\).) Thus one can define \(P^{(\infty)}\) as the
intersection of all \(P^{(n)}\) for finite \(n\). This was
Cantor's first encounter with transfinite iterations.



Then, in late 1873, came a surprising discovery that fully opened the
realm of the transfinite. In correspondence with Dedekind (see Ewald
1996, vol. 2), Cantor asked the question whether the infinite sets
\(\mathbf{N}\) of the natural numbers and \(\mathbf{R}\) of
real numbers can be placed in one-to-one correspondence. In reply,
Dedekind offered a surprising proof that the set \(A\) of all
algebraic numbers is denumerable (i.e., there is a one-to-one
correspondence with \(\mathbf{N}\)). A few days later, Cantor was
able to prove that the assumption that \(\mathbf{R}\) is
denumerable leads to a contradiction. To this end, he employed the
Bolzano-Weierstrass principle of completeness mentioned above. Thus he
had shown that there are more elements in \(\mathbf{R}\) than in
\(\mathbf{N}\) or \(\mathbf{Q}\) or \(A\), in the precise
sense that the cardinality of \(\mathbf{R}\) is strictly greater
than that of \(\mathbf{N}\).


## 2. Consolidation



Set theory was beginning to become an essential ingredient of the new
"modern" approach to mathematics. But this viewpoint was
contested, and its consolidation took a rather long time.
Dedekind's algebraic style only began to find followers in the
1890s; David Hilbert was among them. The soil was better prepared for
the modern theories of real functions: Italian, German, French and
British mathematicians contributed during the 1880s. And the new
foundational views were taken up by Peano and his followers, by Frege
to some extent, by Hilbert in the 1890s, and later by Russell.



Meanwhile, Cantor spent the years 1878 to 1885 publishing key works
that helped turn set theory into an autonomous branch of mathematics.
Let's write \(A \equiv B\) in order to express that the two sets
\(A\), \(B\) can be put in one-to-one correspondence (have
the same cardinality). After proving that the irrational numbers can
be put in one-to-one correspondence with \(\mathbf{R}\), and,
surprisingly, that also \(\mathbf{R}^{n} \equiv \mathbf{R}\), Cantor
conjectured in 1878 that any subset of \(\mathbf{R}\) would be
either denumerable \((\equiv \mathbf{N})\) or \(\equiv \mathbf{R}\).
This is the first and weakest form of the celebrated *Continuum
Hypothesis*. During the following years, Cantor explored the world
of point sets, introducing several important topological ideas (e.g.,
perfect set, closed set, isolated set), and arrived at results such as
the Cantor-Bendixson theorem.



A point set \(P\) is *closed* iff its derived set \(P'
\subseteq P\), and *perfect* iff \(P = P'\). The
Cantor-Bendixson theorem then states that a closed point set can be
decomposed into two subsets \(R\) and \(S\), such that
\(R\) is denumerable and \(S\) is perfect (indeed,
\(S\) is the \(a\)th derived set of \(P\),
for a countable ordinal \(a\)). Because of this, closed sets are
said to have the perfect set property. Furthermore, Cantor was able to
prove that perfect sets have the power of the continuum (1884). Both
results implied that the Continuum Hypothesis is valid for all closed
point sets. Many years later, in 1916, Pavel Aleksandrov and Felix
Hausdorff were able to show that the broader class of Borel sets have
the perfect set property too.



His work on points sets led Cantor, in 1882, to conceive of the
*transfinite numbers* (see Ferreiros 1999:
267*ff*). This was a turning point in his research, for from
then onward he studied *abstract* set theory independently of
more specific questions having to do with point sets and their
topology (until the mid-1880s, these questions had been prominent in
his agenda). Subsequently, Cantor focused on the transfinite cardinal
and ordinal numbers, and on general order types, independently of the
topological properties of \(\mathbf{R}\).



The transfinite ordinals were introduced as new numbers in an
important mathematico-philosophical paper of 1883, *Grundlagen
einer allgemeinen Mannigfaltigkeitslehre* (notice that Cantor
still uses Riemann's term Mannigfaltigkeit or
'manifold' to denote sets). Cantor defined them by means
of two "generating principles": the first (1) yields the
successor \(a+1\) for any given number \(a\), while the second (2)
stipulates that there is a number \(b\) which follows immediately
after any given sequence of numbers without a last element. Thus,
after all the finite numbers comes, by (2), the first transfinite
number, \(\omega\) (read: omega); and this is followed by
\(\omega+1\), \(\omega+2\), ..., \(\omega+\omega = \omega \cdot
2\), ..., \(\omega \cdot n\), \(\omega\cdot n +1\), ...,
\(\omega^{2}\), \(\omega^{2}+1\), ..., \(\omega^{\omega}\),
... and so on and on. Whenever a sequence without last element
appears, one can go on and, so to say, jump to a higher stage by
(2).



The introduction of these new numbers seemed like idle speculation to
most of his contemporaries, but for Cantor they served two very
important functions. To this end, he classified the transfinite
ordinals as follows: the "first number class" consisted of
the finite ordinals, the set \(\mathbf{N}\) of natural numbers;
the "second number class" was formed by o and all
numbers following it (including \(\omega^{\omega}\), and many more)
that have *only a denumerable set of predecessors*. This
crucial condition was suggested by the problem of proving the
Cantor-Bendixson theorem (see Ferreiros 1995). On that basis,
Cantor could establish the results that the cardinality of the
"second number class" is greater than that of
\(\mathbf{N}\); and that no intermediate cardinality exists. Thus,
if you write \(\textit{card}(\mathbf{N}) = \aleph\_{0}\) (read: aleph
zero), his theorems justified calling the cardinality of the
"second number class" \(\aleph\_{1}\).



After the second number class comes a "third number class"
(all transfinite ordinals whose set of predecessors has cardinality
\(\aleph\_{1}\)); the cardinality of this new number class can be
proved to be \(\aleph\_{2}\). And so on. The first function of the
transfinite ordinals was, thus, to establish a well-defined scale of
increasing transfinite cardinalities. (The aleph notation used above
was introduced by Cantor only in 1895.) This made it possible to
formulate much more precisely the problem of the continuum;
Cantor's conjecture became the hypothesis that
\(\textit{card}(\mathbf{R}) = \aleph\_{1}\). Furthermore, relying on
the transfinite ordinals, Cantor was able to prove the
Cantor-Bendixson theorem, rounding out the results on point sets that
he had been elaborating during these crucial years. The
Cantor-Bendixson theorem states: closed sets of \(\mathbf{R}^n\)
(generalizable to Polish spaces) have the perfect set property, so
that any closed set \(S\) in \(\mathbf{R}^n\) can be written uniquely
as the disjoint union of a perfect set \(P\) and a countable set
\(R\). Moreover, \(P\) is \(S^a\) for a countable
ordinal.



The study of the transfinite ordinals directed Cantor's
attention towards ordered sets, and in particular *well-ordered
sets*. A set \(S\) is well-ordered by a relation < iff
< is a total order and every subset of \(S\) has a least
element in the <-ordering. (The real numbers are not well-ordered
in their usual order: just consider an open interval. Meanwhile,
\(\mathbf{N}\) is the simplest infinite well-ordered set.) Cantor
argued that the transfinite ordinals truly deserve the name of
*numbers*, because they express the "type of order"
of any possible well-ordered set. Notice also that it was easy for
Cantor to indicate how to reorder the natural numbers so as to make
them correspond to the order types \(\omega+1\), \(\omega+2\),
..., \(\omega \cdot 2\), ..., \(\omega \cdot n\), ...,
\(\omega^2\), ..., \(\omega^{\omega}\), ... and so on. (For
instance, reordering \(\mathbf{N}\) in the form: 2, 4, 6,
..., 5, 15, 25, 35, ..., 1, 3, 7, 9, ... we obtain a
set that has order type \(\omega\cdot 3\).)



Notice too that the Continuum Hypothesis, if true, would entail that
the set \(\mathbf{R}\) of real numbers can indeed be well-ordered.
Cantor was so committed to this viewpoint, that he presented the
further hypothesis that *every* set can be well-ordered as
"a fundamental and momentous law of thought". Some years
later, Hilbert called attention to both the Continuum Hypothesis and
the well-ordering problem as Problem 1 in his celebrated list of
'Mathematische Probleme' (1900). Doing so was an
intelligent way of emphasizing the importance of set theory for the
future of mathematics, and the fruitfulness of its new methods and
problems.



In 1895 and 1897, Cantor published his last two articles. They were a
well-organized presentation of his results on the transfinite numbers
(cardinals and ordinals) and their theory, and also on order types and
well-ordered sets. However, these papers did not advance significant
new ideas. Unfortunately, Cantor had doubts about a third part he had
prepared, which would have discussed very important issues having to
do with the problem of well-ordering and the paradoxes (see below).
Surprisingly, Cantor also failed to include in the 1895/97 papers a
theorem which he had published some years before which is known simply
as Cantor's Theorem: given any set \(S\), there exists another
set whose cardinality is greater (this is the power set
\(\mathcal{P}(S)\), as we now say--Cantor used instead the set of
all functions of the form \(f\): \(S \rightarrow \{0, 1\}\), which is
equivalent). In the same short paper (1892), Cantor presented his
famous proof that \(\mathbf{R}\) is non-denumerable by the method of
diagonalisation, a method which he then extended to prove
Cantor's Theorem. (A related form of argument had appeared
earlier in the work of P. du Bois-Reymond [1875], see among others
[Wang 1974, 570] and [Borel 1898], Note II.)



Meanwhile, other authors were exploring the possibilities opened by
set theory for the foundations of mathematics. Most important was
Dedekind's contribution (1888) with a deep presentation of the
theory of the natural numbers. He formulated some basic principles of
set (and mapping) theory; gave axioms for the natural number system;
proved that mathematical induction is conclusive and recursive
definitions are flawless; developed the basic theory of arithmetic;
introduced the finite cardinals; and proved that his axiom system is
categorical. His system had four axioms. Given a function ph
defined on \(S\), a set \(N \subseteq S\), and a distinguished
element \(1 \in N\), they are as follows:

 

\[
\begin{align}
\tag{a} & \phi(N) \subset N\\
\tag{b} & N = \phi\_{o}\{1\}\\
\tag{g} & 1 \not\in \phi(N)\\
\tag{d} & \textrm{the function } \phi \textrm{ is injective.}
\end{align}
\]


Condition (b) is crucial since it ensures minimality for the set
of natural numbers, which accounts for the validity of proofs by
mathematical induction. \(N = \phi\_{o}\{1\}\) is read: \(N\) is
the *chain of* singleton {1} under the function ph, that is,
the minimal closure of {1} under the function ph. In general, one
considers the chain of a set \(A\) under an arbitrary mapping
g, denoted by \(\gamma\_{o}(A)\); in his booklet Dedekind
developed an interesting theory of such chains, which allowed him to
prove the Cantor-Bernstein theorem. The theory was later generalized
by Zermelo and applied by Skolem, Kuratowski, etc.



In the following years, Giuseppe Peano gave a more superficial (but
also more famous) treatment of the natural numbers, employing the new
symbolic language of logic, and Gottlob Frege elaborated his own
ideas, which however fell prey to the paradoxes. An important book
inspired by the set-theoretic style of thinking was Hilbert's
*Grundlagen der Geometrie* (1899), which took the
"mathematics of axioms" one step beyond Dedekind through a
rich study of geometric systems motivated by questions concerning the
independence of his axioms. Hilbert's book made clear the new
axiomatic methodology that had been shaping up in connection with the
novel methods of set theory, and he combined it with the axiomatic
trends coming from projective geometry.



Nevertheless, as we said before, there was quite a lot of criticism of
set-theoretic, infinitarian methods. As early as 1870, Kronecker had
begun to voice critical remarks of a constructivist bent that, many
years later, would be echoed by prominent thinkers like Brouwer or
Wittgenstein. Kronecker's critical orientation pointed in the
way of renouncing the real number system and classical analysis, in
favor of some more stringent form of analysis -- twentieth
century examples of this would be predicative analysis (H. Weyl
building on basic notions of Poincare, see Feferman 1988) and
intuitionistic analysis (Brouwer). Even Weierstrass had objections (in
1874, at least) against the idea of distinguishing sizes of infinity,
and that on the face of Cantor's proofs. Examples abound, and so
during the 1900s many mathematicians expressed doubts about key ideas
and methods of set theory. A prototype case is E. Borel, who after
introducing the ideas of Cantor in France [1898], became increasingly
suspicious of set theory (the five letters exchanged by him and Baire,
Lebesgue, Hadamard in 1905 have become famous; see Ewald [1996,
vol. 2]). But there are also the cases of Poincare, Weyl,
Skolem, and so on. Among philosophers, the most prominent example is
Wittgenstein, who condemned set theory for building on the "nonsense"
of fictitious symbolism, suggesting "wrong imagery", and
so on.
 


## 3. Critical Period



In the late nineteenth century, it was a widespread idea that pure
mathematics is nothing but an elaborate form of arithmetic. Thus it
was usual to talk about the "*arithmetisation*" of
mathematics, and how it had brought about the highest standards of
rigor. With Dedekind and Hilbert, this viewpoint led to the idea of
*grounding all of pure mathematics in set theory*. The most
difficult steps in bringing forth this viewpoint had been the
establishment of a theory of the real numbers, and a set-theoretic
reduction of the natural numbers. Both problems had been solved by the
work of Cantor and Dedekind. But precisely when mathematicians were
celebrating that "full rigor" had been finally attained,
serious problems emerged for the foundations of set theory. First
Cantor, and then Russell, discovered the paradoxes in set theory.



Cantor was led to the paradoxes by having introduced the
"conceptual sphere" of the transfinite numbers. Each
transfinite ordinal is the order type of the set of its predecessors;
e.g., o is the order type of \(\{0, 1, 2, 3, \ldots\}\), and
\(\omega+2\) is the order type of \(\{0, 1, 2, 3, \ldots, \omega,
\omega +1\}\). Thus, to each initial segment of the series of
ordinals, there corresponds an immediately greater ordinal. Now, the
"whole series" of all transfinite ordinals would form a
well-ordered set, and to it there would correspond a new ordinal
number. This is unacceptable, for this ordinal \(o\) would have
to be greater than all members of the "whole series", and
in particular \(o < o\). This is usually called the
*Burali-Forti paradox*, or paradox of the ordinals (although
Burali-Forti himself failed to formulate it clearly, see Moore &
Garciadiego 1981).



Although it is conceivable that Cantor might have found that paradox
as early as 1883, immediately after introducing the transfinite
ordinals (for arguments in favour of this idea see Purkert &
Ilgauds 1987 and Tait 2000), the evidence indicates clearly that it
was not until 1896/97 that he found this paradoxical argument and
realized its implications. By this time, he was also able to employ
Cantor's Theorem to yield the
*Cantor paradox*, or paradox of the alephs: if there existed a
"set of all" cardinal numbers (alephs), Cantor's
Theorem applied to it would give a new aleph \(\aleph\), such that
\(\aleph < \aleph\). The great set theorist realized perfectly well
that these paradoxes were a fatal blow to the "logical"
approaches to sets favoured by Frege and Dedekind. Cantor emphasized
that his views were "*in diametrical opposition*"
to Dedekind's, and in particular to his "naive
assumption that *all well-defined collections, or systems*, are
also '*consistent systems*' " (see the letter
to Hilbert, Nov. 15, 1899, in Purkert & Ilgauds 1987: 154).
(Contrary to what has often been claimed, Cantor's ambiguous
definition of set in his paper of 1895 was intended to be
"diametrically opposite" to the logicists'
understanding of sets--often called "naive" set
theory, which could more properly be called the dichotomy conception of
sets, following a suggestion of Godel.)



Cantor thought he could solve the problem of the paradoxes by
distinguishing between "consistent multiplicities" or
sets, and "inconsistent multiplicities". But, in the
absence of explicit criteria for the distinction, this was simply a
verbal answer to the problem. Being aware of deficiencies in his new
ideas, Cantor never published a last paper he had been preparing, in
which he planned to discuss the paradoxes and the problem of
well-ordering (we know quite well the contents of this unpublished
paper, as Cantor discussed it in correspondence with Dedekind and
Hilbert; see the 1899 letters to Dedekind in Cantor 1932, or Ewald
1996: vol. 2). Cantor presented an argument that relied on the
"Burali-Forti" paradox of the ordinals, and aimed to prove
that every set can be well-ordered. This argument was later
rediscovered by the British mathematician P.E.B. Jourdain, but it is
open to criticism because it works with "inconsistent
multiplicities" (Cantor's term in the above-mentioned
letters).



Cantor's paradoxes convinced Hilbert and Dedekind that there
were important doubts concerning the foundations of set theory.
Hilbert formulated a paradox of his own (Peckhaus & Kahle 2002),
and discussed the problem with mathematicians in his Gottingen
circle. Ernst Zermelo was thus led to discover the paradox of the
"set" of all sets that are not members of themselves (Rang
& Thomas 1981). This was independently discovered by Bertrand
Russell, who was led to it by a careful study of Cantor's
Theorem, which conflicted deeply with Russell's belief in a
universal set. Some time later, in June 1902, he communicated the
"contradiction" to Gottlob Frege, who was completing his
own logical foundation of arithmetic, in a well-known letter [van
Heijenoort 1967, 124]. Frege's reaction made very clear the
profound impact of this contradiction upon the logicist program.
"Can I always speak of a class, of the extension of a concept?
And if not, how can I know the exceptions?" Faced with this,
"I cannot see how arithmetic could be given a scientific
foundation, how numbers could be conceived as logical objects"
(Frege 1903: 253).



The publication of Volume II of Frege's *Grundgesetze*
(1903), and above all Russell's work *The Principles of
Mathematics* (1903), made the mathematical community fully aware
of the existence of the set-theoretic paradoxes, of their impact and
importance. There is evidence that, up to then, even Hilbert and
Zermelo had not fully appreciated the damage. Notice that the
Russell-Zermelo paradox operates with very basic
notions--negation and set membership--concepts that had
widely been regarded as purely logical. The "set" \(R =
\{x: x \not\in x\}\) exists according to the principle of
comprehension (which allows any open sentence to determine a class),
but if so, \(R \in R \textit{ iff } R \not\in R\). It is a direct
contradiction to the principle favoured by Frege and Russell.



It was obviously necessary to clarify the foundations of set theory,
but the overall situation did not make this an easy task. The
different competing viewpoints were widely divergent. Cantor had a
metaphysical understanding of set theory and, although he had one of
the sharpest views of the field, he could not offer a precise
foundation. It was clear to him (as it had been, somewhat
mysteriously, to Ernst Schroder in his *Vorlesungen uber
die Algebra der Logik*, 1891) that one has to reject the idea of a
Universal Set, favoured by Frege and Dedekind. Frege and Russell based
their approach on the principle of comprehension, which was shown
contradictory. Dedekind avoided that principle, but he postulated that
the Absolute Universe was a set, a "thing" in his
technical sense of *Gedankending*; and he coupled that
assumption with full acceptance of arbitrary subsets.



This idea of admitting arbitrary subsets had been one of the deep
inspirations of both Cantor and Dedekind, but none of them had
thematized it. (Here, their modern understanding of analysis played a
crucial but implicit background role, since they worked within the
Dirichlet-Riemann tradition of "arbitrary" functions.) As
for the now famous iterative conception there were some elements of it
(particularly in Dedekind's work, with his iterative development
of the number system, and his views on "systems" and
"things"), but it was conspicuously absent from many of
the relevant authors. Typically, e. g., Cantor did not iterate the
process of set formation: he tended to consider sets of
*homogeneous* elements, elements which were taken to belong
"in some conceptual sphere" (either numbers, or points, or
functions, or even physical particles--but not intermingled). The
iterative conception was first suggested by Kurt Godel in [1933],
in connection with technical work by von Neumann and Zermelo a few
years earlier; Godel would insist on the idea in his well-known
paper on Cantor's continuum problem. It came only *post
facto*, after very substantial amounts of set theory had been
developed and fully systematized.



This variety of conflicting viewpoints contributed much to the overall
confusion, but there was more. In addition to the paradoxes discussed
above (set-theoretic paradoxes, as we say), the list of
"logical" paradoxes included a whole array of further ones
(later called "semantic"). Among these are paradoxes due
to Russell, Richard, Konig, Berry, Grelling, etc., as well as the
ancient liar paradox due to Epimenides. And the diagnoses and proposed
cures for the damage were tremendously varied. Some authors, like
Russell, thought it was essential to find a new logical system that
could solve all the paradoxes at once. This led him into the ramified
type theory that formed the basis of *Principia Mathematica* (3
volumes, Whitehead and Russell 1910-1913), his joint work with Alfred
Whitehead. Other authors, like Zermelo, believed that most of those
paradoxes dissolved as soon as one worked within a restricted
axiomatic system. They concentrated on the "set-theoretic"
paradoxes (as we have done above), and were led to search for
axiomatic systems of set theory.



Even more importantly, the questions left open by Cantor and
emphasized by Hilbert in his first problem of 1900 caused heated
debate. At the International Congress of Mathematicians at Heidelberg,
1904, Gyula (Julius) Konig proposed a very detailed proof that
the cardinality of the continuum *cannot be* any of
Cantor's alephs. His proof was only flawed because he had relied
on a result previously "proven" by Felix Bernstein, a
student of Cantor and Hilbert. It took some months for Felix Hausdorff
to identify the flaw and correct it by properly stating the special
conditions under which Bernstein's result was valid (see
Hausdorff 2001, vol. 1). Once thus corrected, Konig's
theorem became one of the very few results restricting the possible
solutions of the continuum problem, implying, e.g., that
\(\textit{card}(\mathbf{R})\) is not equal to \(\aleph\_{\omega}\).
Meanwhile, Zermelo was able to present a proof that every set can be
well-ordered, using the Axiom of Choice [1904]. During the following
year, prominent mathematicians in Germany, France, Italy and England
discussed the Axiom of Choice and its acceptability.



The Axiom of Choice states: For every set \(A\) of non-empty sets, there exists a set that has exactly one element in common with each set in \(A\).
This started a whole era during which the Axiom of Choice was treated
most carefully as a dubious hypothesis (see the monumental study by
Moore 1982). And that is ironic, for, among all of the usual
principles of set theory, the Axiom of Choice is the only one that
explicitly enforces the existence of some arbitrary subsets. But,
important as this idea had been in motivating Cantor and Dedekind, and
however entangled it is with classical analysis, infinite arbitrary
subsets were rejected by many other authors. Among the most
influential ones in the following period, one ought to emphasize the
names of Russell, Hermann Weyl, and of course Brouwer.


 Choice was, for a long time, a controversial axiom. On the one
hand, it is of wide use in mathematics and, indeed, it's key to
many important theorems of analysis (this became gradually clear with
works such as Sierpinski [1918]). On the other hand, it has
rather unintuitive consequences, such as the Banach-Tarski Paradox,
which says that the unit ball can be partitioned into finitely-many
'pieces' (subsets), which can then be rearranged to form
two unit balls (see Tomkowicz & Wagon [2019]). The objections to
the axiom arise from the fact that it asserts the existence of sets
that cannot be explicitly defined. During the 1920s and 1930s, there
existed the ritual practice of mentioning it explicitly, whenever a
theorem would depend on the axiom. This stopped only after
Godel's proof of relative consistency, discussed
below. 



The impressive polemics which surrounded his Well-Ordering Theorem,
and the most interesting and difficult problem posed by the
foundations of mathematics, led Zermelo to concentrate on axiomatic
set theory. As a result of his incisive analysis, in 1908 he published
his axiom system, showing how it blocked the known paradoxes and yet
allowed for a masterful development of the theory of cardinals and
ordinals. This, however, is the topic of the entry
 Zermelo's axiomatization of set theory; 
also, on the life and work of Zermelo, see Ebbinghaus 2015.


## 4. From Zermelo to Godel



In the period 1900-1930, the rubric "set theory" was
still understood to include topics in topology and the theory of
functions. Although Cantor, Dedekind, and Zermelo had left that stage
behind to concentrate on pure set theory, for mathematicians at large
this would still take a long time. Thus, at the first International
Congress of Mathematicians, 1897, keynote speeches given by Hadamard
and Hurwitz defended set theory on the basis of its importance for
analysis. Around 1900, motivated by topics in analysis, important work
was done by three French experts: Borel [1898], Baire [1899] and
Lebesgue [1902] [1905]. Their work inaugurated the development of
descriptive set theory by extending Cantor's studies on
definable sets of real numbers (in which he had established that the
Continuum Hypothesis is valid for closed sets). They introduced the
hierarchy of Borel sets, the Baire hierarchy of functions, and the
concept of Lebesgue measure--a crucial concept of modern
analysis.



Descriptive set theory (DST) is the study of certain kinds of
definable sets of real numbers, which are obtained from simple kinds
(like the open sets and the closed sets) by well-understood operations
like complementation or projection. The *Borel sets* were the
first hierarchy of definable sets, introduced in the 1898 book of
Emile Borel; they are obtained from the open sets by iterated
application of the operations of countable union and complementation.
In 1905 Lebesgue studied the Borel sets in an epochal memoir, showing
that their hierarchy has levels for all countable ordinals, and
analyzing the Baire functions as counterparts of the Borel sets. The
main aim of descriptive set theory is to find structural properties
common to all such definable sets: for instance, the Borel sets were
shown to have the perfect set property (if uncountable, they have a
perfect subset) and thus to comply with the continuum hypothesis (CH).
This result was established in 1916 by Hausdorff and by Alexandroff,
working independently. Other important "regularity
properties" studied in DST are the property of being Lebesgue
measurable, and the so-called property of Baire (to differ from an
open set by a so-called meager set, or set of first category).



Also crucial at the time was the study of the *analytic* sets,
namely the continuous images of Borel sets, or equivalently, the
projections of Borel sets. The young Russian mathematician Mikhail
Suslin found a mistake in Lebesgue's 1905 memoir when he
realized that the projection of a Borel set is not Borel in general
[Suslin 1917]. However, he was able to establish that the analytic
sets, too, possess the perfect set property and thus verify CH. By
1923 Nikolai Lusin and Waclaw Sierpinski were studying the
*co-analytic* sets, and this was to lead them to a new
hierarchy of *projective* sets, which starts with the analytic
sets \((\Sigma^{1}\_{1})\), their complements (co-analytic,
\(\Pi^{1}\_{1}\) sets), the projections of these last
(\(\Sigma^{1}\_{2}\) sets), their complements (\(\Pi^{1}\_{2}\) sets),
and so on. During the 1920s much work was done on these new types of
sets, mainly by Polish mathematicians around Sierpinski and by
the Russian school of Lusin and his students. A crucial result
obtained by Sierpinski was that every \(\Sigma^{1}\_{2}\) set is the
union of \(\aleph\_{1}\) Borel sets (the same holds for
\(\Sigma^{1}\_{1}\) sets), but this kind of traditional research on the
topic would stagnate after around 1940 (see Kanamori [1995]).



Soon Lusin, Sierpinski and their colleagues were finding extreme
difficulties in their work. Lusin was so much in despair that, in a
paper of 1925, he came to
the "totally unexpected" conclusion that "one does
not know and one will never know" whether the projective sets
have the desired regularity properties (quoted in Kanamori 1995: 250).
Such comments are highly interesting in the light of later
developments, which have led to hypotheses that solve all the relevant
questions (Projective Determinacy, in particular). They underscore the difficult
methodological and philosophical issues raised by these more recent
hypotheses, namely the problem concerning the kind of evidence that
backs them.



Lusin summarized the state of the art in his 1930 book
*Lecons sur les ensembles analytiques* (Paris,
Gauthier-Villars), which was to be a key reference for years to come.
Since this work, it has become customary to present results in DST for
the Baire space \( ^{\omega}\)\(\omega\) of infinite sequences of natural
numbers, which in effect had been introduced by Rene Baire in a
paper published in 1909. Baire space is endowed with a certain
topology that makes it homeomorphic to the set of the irrational
numbers, and it is regarded by experts to be "perhaps the most
fundamental object of study of set theory" next to the set of
natural numbers [Moschovakis 1994, 135].



This stream of work on DST must be counted among the most important
contributions made by set theory to analysis and topology. But what
had begun as an attempt to prove the Continuum Hypothesis could not
reach this goal. Soon it was shown using the Axiom of Choice that
there are non-Lebesgue measurable sets of reals (Vitali 1905), and
also uncountable sets of reals with no perfect subset (Bernstein
1908). Such results made clear the impossibility of reaching the goal
of CH by concentrating on definable and "well-behaved"
sets of reals.



Also, with Godel's work around 1940 (and also with forcing
in the 1960s) it became clear why the research of the 1920s and 30s
had stagnated: the fundamental new independence results showed that
the theorems established by Suslin (perfect set property for analytic
sets), Sierpinski (\(\Sigma^{1}\_{2}\) sets as unions of \(\aleph\_{1}\)
Borel sets) and a few others were the best possible results on the
basis of axiom system ZFC. This is important philosophically: already
an exploration of the world of sets definable from the open (or
closed) sets by complement, countable union, and projection had
sufficed to reach the limits of the ZFC system. Hence the need for new
axioms, that Godel emphasized after World War II [Godel
1947].



Let us now turn to Cantor's other main legacy, the study of
transfinite numbers. By 1908 Hausdorff was working on uncountable
order types and introduced the Generalized Continuum Hypothesis
\((2^{\aleph\_{a}} = \aleph\_{a+1})\). He was also the first to consider
the possibility of an "exorbitant" cardinal, namely a
weakly inaccessible, i.e., a regular cardinal that is not a successor
(a cardinal \(\alpha\) is called regular if decomposing \(\alpha\)
into a sum of smaller cardinals requires \(\alpha\)-many such
numbers). Few years later, in the early 1910s, Paul Mahlo was studying
hierarchies of such large cardinals in work that pioneered what was to
become a central area of set theory; he obtained a succession of
inaccessible cardinals by employing a certain operation that involves
the notion of a stationary subset; they are called Mahlo cardinals.
But the study of large cardinals developed slowly. Meanwhile,
Hausdorff's textbook *Grundzuge der Mengenlehre*
(1914) introduced two generations of mathematicians into set theory
and general topology.



The next crucial steps into the "very high" infinite were
done in 1930. The notion of strongly inaccessible cardinals was then
isolated by Sierpinski & Tarski, and by Zermelo [1930]. A
strong inaccessible is a regular cardinal \(\alpha\) such that \(2^x\)
is less than \(\alpha\) whenever \(x < \alpha\). While weak
inaccessibles merely involve closure under the successor operation,
strong inaccessibles involve a much stronger notion of closure under
the powerset operation. That same year, in a path-breaking paper on
models of ZFC, Zermelo [1930] established a link between the
uncountable (strongly) inaccessible cardinals and certain
"natural" models of ZFC (in which work he assumed that the
powerset operation is, so to say, fully determinate).



In that same year, Stanislaw Ulam was led by considerations coming out
of analysis (measure theory) to a concept that was to become central:
measurable cardinals. It turned out that such cardinals, defined by a
measure-theoretic property, had to be (strongly) inaccessible. Indeed,
many years later it would be established (by Hanf, working upon
Tarski's earlier work) that the first inaccessible cardinal is
not measurable, showing that these new cardinals were even more
"exorbitant". As one can see, the Polish school led by
Sierpinski had a very central role in the development of set
theory between the Wars. Measurable cardinals came to special
prominence in the late 1960s when it became clear that the existence
of a measurable cardinal contradicts Godel's axiom of
constructibility (\(V = L\) in the class notation). This again
vindicated Godel's convictions, expressed in what is
sometimes called "Godel's program" for new
axioms.



Set-theoretic mathematics continued its development into the powerful
axiomatic and structural approach that was to dominate much of the
20th century. To give just a couple of examples,
Hilbert's early axiomatic work (e.g., in his arch-famous
*Foundations of Geometry*) was deeply set-theoretic; Ernst
Steinitz published in 1910 his research on abstract field theory,
making essential use of the Axiom of Choice; and around the same time
the study of function spaces began with work by Hilbert, Maurice
Frechet, and others. During the 1920s and 30s, the first
specialized mathematics journal, *Fundamenta Mathematicae*, was
devoted to set theory as then understood (centrally including topology
and function theory). In those decades structural algebra came of age,
abstract topology was gradually becoming an independent branch of
study, and the study of set theory initiated its metatheoretic turn.



Ever since, "set theory" has generally been identified
with the branch of mathematical logic that studies transfinite sets,
originating in Cantor's result that \(\mathbf{R}\) has a
greater cardinality than \(\mathbf{N}\). But, as the foregoing
discussion shows, set theory was both effect and cause of the rise of
modern mathematics: the traces of this origin are indelibly stamped on
its axiomatic structure.