Set theory is one of the greatest achievements of modern mathematics.Basically all mathematical concepts, methods, and results admit ofrepresentation within axiomatic set theory. Thus set theory has servedquite a unique role by systematizing modern mathematics, andapproaching in a unified form all basic questions about admissiblemathematical arguments—including the thorny question ofexistence principles. This entry covers in outline the convolutedprocess by which set theory came into being, covering roughly theyears 1850 to 1930.

In 1910, Hilbert wrote that set theory is

that mathematical discipline which today occupies an outstanding rolein our science, and radiates [ausströmt] its powerfulinfluence into all branches of mathematics. [Hilbert 1910, 466;translation by entry author]

This already suggests that, in order to discuss the early history, itis necessary to distinguish two aspects of set theory: its role as afundamental language and repository of the basic principles of modernmathematics; and its role as an independent branch of mathematics,classified (today) as a branch of mathematical logic. Both aspects areconsidered here.

The first section examines the origins andemergence of settheoretic mathematics around 1870; this is followed by a discussion ofthe period of expansion andconsolidation of the theory up to1900. Section 3 provides a look at thecritical period in thedecades 1897 to 1918, and Section 4 deals with the time from Zermeloto Gödel (from theory to metatheory), with special attention tothe often overlooked, but crucial,descriptive set theory.

• 1. Emergence
• 2. Consolidation
• 3. Critical Period
• 4. From Zermelo to Gödel
• Bibliography
  • Cited Works
  • Further Reading
• Academic Tools
• Other Internet Resources
• Related Entries

1. Emergence

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

1. It is not the case that actual infinity was universally rejectedbefore Cantor.
2. Set-theoretic views did not arise exclusively from analysis, butemerged also in algebra, number theory, and geometry.
3. In fact, the rise of set-theoretic mathematics precededCantor’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 ideasabout classes have existed since at least the “tree ofPorphyry” (3^rd century CE). Thus it becomes difficultto sort out the origins of the concept of set. But sets are neithercollections in the everyday sense of this word, nor“classes” or concept-extensions in the sense of logiciansbefore the mid-19^th century. The key missing element isobjecthood — a set is a mathematical object, to be operated uponjust like any other object (the set \(\mathbf{N}\) is as much ‘athing’ as number 3). To clarify this point, Russell employed theuseful distinction between a class-as-many (this is closer to thetraditional idea) and a class-as-one (or set).

The idea of a concept-extension prepared the way for the modern notionof set, but there are still significant differences. As handled bylogicians of the seventeenth, eighteenth, and early-nineteenthcentury, the extension of a concept is “the ordered totality ofall the kinds [Arten] subordinated to the concept”, i.e. thecollection of all concepts that fall under it (quoting from theLogik of Drobisch, 1851). Notice that there is no referenceto objects or individuals as the elements of the class, since theelements that form the extension are again concepts. One can thennotice two key changes involved in the absorption of the idea ofconcept-extension by mathematicians: the key simplification ofconsidering classes of individuals, and the extreme generalizationinvolved in expanding beyond any particular “conceptualsphere”, towards absolute generality.

In any event, since those differences are not so obvious, the linkbetween sets and (old) classes prepared the way for the rise oflogicism. The assumptions of the centrality of concepts in logic, andthe immediate connection between concepts and classes, were deeplyingrained in the minds of logicians; both can be followed up until thevery end of the century (e.g., see Ferreirós 2009). The second,in particular, — with the provisos indicated in the previousparagraph — is the root of the principle of comprehension thatplayed such a crucial role around 1900.

Ernst Zermelo, a crucial figure in our story, said that the theory hadhistorically been “created by Cantor and Dedekind”[Zermelo 1908, 262]. This suggests a good pragmatic criterion foranalyzing the early history: one should start from authors who havesignificantly influenced the conceptions of Cantor, Dedekind, andZermelo. For the most part, this is the criterion adopted here.Nevertheless, as every rule calls for an exception, the case ofBolzano is important and instructive, even though Bolzano did notsignificantly influence later writers.

In 19^th century German-speaking areas, there were someintellectual tendencies that promoted the acceptance of the actualinfinite (e.g., a revival of Leibniz’s thought). In spite ofGauss’s warning that the infinite can only be a manner ofspeaking, some minor figures and three major ones (Bolzano, Riemann,Dedekind) preceded Cantor in fully accepting the actual infinite inmathematics. Those three authors were active in promoting aset-theoretic reformulation of mathematical ideas, withDedekind’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 almostno influence. The high quality of his work in logic and thefoundations of mathematics is well known. A book entitledParadoxien des Unendlichen was posthumously published in1851. Here Bolzano argued in detail that a host of paradoxessurrounding infinity are logically harmless, and mounted a forcefuldefence of actual infinity. He proposed an interesting argumentattempting to prove the existence of infinite sets, which bearscomparison with Dedekind’s later argument (1888). Although heemployed complicated distinctions of different kinds of sets orclasses, Bolzano recognized clearly the possibility of putting twoinfinite 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 setsare “equal with respect to the multiplicity of theirparts” [1851, 30–31]. In all likelihood, traditional ideasof measurement were still too powerful in his way of thinking, andthus 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 geometryand especially of topology) was to have a crucial role in makingpossible the abstract viewpoint of set theory. Bernhard Riemannproposed visionary ideas about topology, and about basing all ofmathematics on the notion of set or “manifold” in thesense of class (Mannigfaltigkeit), in his celebratedinaugural lecture “On the Hypotheses which lie at theFoundations of Geometry” (1854/1868a). Also characteristic ofRiemann was a great emphasis onconceptual mathematics,particularly visible in his approach to complex analysis (which againwent deep into topology). To give but the simplest example, Riemannwas an enthusiastic follower of Dirichlet’s idea that a functionhas to be conceived as an arbitrary correspondence between numericalvalues, be it representable by a formula or not; this meant leavingbehind the times when a function was defined to be an analyticexpression. Through this new style of mathematics, and through hisvision of a new role for sets and a full program for developingtopology, Riemann was a crucial influence on both Dedekind and Cantor(see Ferreirós 1999).

The five-year period 1868–1872 saw a mushrooming ofset-theoretic proposals in Germany, so much so that we could regard itas the birth of set-theoretic mathematics. Riemann’s geometrylecture, delivered in 1854, was published by Dedekind in 1868, jointlywith Riemann’s paper on trigonometric series (1854/1868b, whichpresented the Riemann integral). The latter was the starting point fordeep work in real analysis, commencing the study of“seriously” discontinuous functions. The young GeorgCantor entered into this area, which led him to the study ofpoint-sets. In 1872 Cantor introduced an operation upon point sets(see below) and soon he was ruminating about the possibility toiterate that operation to infinity and beyond: it was the firstglimpse of the transfinite realm.

Meanwhile, another major development had been put forward by RichardDedekind in 1871. In the context of his work on algebraic numbertheory, Dedekind introduced an essentially set-theoretic viewpoint,defining fields and ideals of algebraic numbers. These ideas werepresented in a very mature form, making use of set operations and ofstructure-preserving mappings (see a relevant passage inFerreirós 1999: 92–93; Cantor employed Dedekind’sterminology for the operations in his own work on set theory as lateas 1880 [1999: 204]). Considering the ring of integers in a givenfield of algebraic numbers, Dedekind defined certain subsets called“ideals” and operated on these sets as new objects. Thisprocedure was the key to his general approach to the topic. In otherworks, he dealt very clearly and precisely with equivalence relations,partition sets, homomorphisms, and automorphisms (on the history ofequivalence relations, see Asghari ‎2018). Thus, many of the usualset-theoretic procedures of twentieth-century mathematics go back tohis work. Several years later (in 1888), Dedekind would publish apresentation of the basic elements of set theory, making a bit moreexplicit the operations on sets and mappings he had been using since1871.

The following year, Dedekind published a paper [1872] in which heprovided an axiomatic analysis of the structure of the set\(\mathbf{R}\) of real numbers. He defined it as an ordered field thatis 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, employingCauchy sequences of rational numbers, which was an elegantsimplification of the definition offered by Carl Weierstrass in hislectures. The form of completeness axiom that Weierstrass preferredwas Bolzano’s principle that a sequence of nested closedintervals in \(\mathbf{R}\) (a sequence such that \([a_{m+1},b_{m+1}]\subset [a_{m},b_{m}]\)) “contains” at least one realnumber (or, as we would say, has a non-empty intersection).

The Cantor and Dedekind definitions of the real numbers reliedimplicitly on set theory, and can be seen in retrospect to involve theassumption of a Power Set principle. Both took as given the set ofrational numbers, and for the definition of \(\mathbf{R}\) they reliedon a certain totality of infinite sets of rational numbers (either thetotality of Cauchy sequences, or of all Dedekind cuts). In reaction tothis, constructivistic criticism of set theory began to emerge, asLeopold Kronecker started to make objections to such infinitaryprocedures. 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 severalauthors in the fields of real analysis and complex analysis (e.g.,Hankel, du Bois-Reymond, H.J.S. Smith, U. Dini) and by Dedekind injoint work with Weber (1882), pioneering algebraic geometry.

Cantor’s derived sets are of particular interest since they ledCantor to start considering transfinite iterations (for the context ofthis idea in real analysis, see e.g., Dauben 1979, Hallett 1984,Lavine 1994, Kanamori 1996, Ferreirós 1999). Cantor took asgiven the “conceptual sphere” of the real numbers, and heconsidered arbitrary subsets \(P\), which he called ‘pointsets’. A real number \(r\) is called alimit point of\(P\), when all neighbourhoods of \(r\) contain points of \(P\). Thiscan only happen if \(P\) is infinite. With that concept, due toWeierstrass, Cantor went on to define thederived set \(P'\)of \(P\), as theset of all the limit points of \(P\). Ingeneral \(P'\) may be infinite and have its own limit points (seeCantor’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 giveexamples 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 unitinterval; 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 transfiniteiterations.

Then, in late 1873, came a surprising discovery that fully opened therealm of the transfinite. In correspondence with Dedekind (see Ewald1996, vol. 2), Cantor asked the question whether the infinite sets\(\mathbf{N}\) of the natural numbers and \(\mathbf{R}\) of realnumbers can be placed in one-to-one correspondence. In reply, Dedekindoffered a surprising proof that the set \(A\) of all algebraic numbersis denumerable (i.e., there is a one-to-one correspondence with\(\mathbf{N}\)). A few days later, Cantor was able to prove that theassumption that \(\mathbf{R}\) is denumerable leads to acontradiction. To this end, he employed the Bolzano-Weierstrassprinciple of completeness mentioned above. Thus he had shown thatthere 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 \(\mathbf{N}\).

All of these results appeared in an 1874 paper, ‘On a propertyof the collection of all real algebraic numbers’, that is justlyregarded as the birth of transfinite set theory. Unfortunately, thispublication also led to serious difficulties in the relations betweenCantor and Dedekind, due to the way in which Cantor employed theletters and ideas of his partner (see Ferreirós 1993); S.Müller-Stach (2024) talks of Cantor “thus violatingscientific standards boldly”, while O. Deiser (2020) speaks infavor of Cantor. This unpleasant episode was somehow related to the“Berlin circumstances”, and indeed Weierstraß hadgreat interest in the theorem establishing a simple, well-orderedsequence of algebraic numbers (which is highlighted in thepaper’s title, and was contributed by Dedekind). To us, whatseems particularly relevant is the second theorem, which isundoubtedly Cantor’s own.

2. Consolidation

Set theory was beginning to become an essential ingredient of the new“modern” approach to mathematics. But this viewpoint wascontested, and its consolidation took a rather long time.Dedekind’s algebraic style only began to find followers in the1890s; David Hilbert was among them. The soil was better prepared forthe modern theories of real functions: Italian, German, French andBritish mathematicians contributed during the 1880s. And the newfoundational views were taken up by Peano and his followers, by Fregeto some extent, by Hilbert in the 1890s, and later by Russell.

Meanwhile, Cantor spent the years 1878 to 1885 publishing key worksthat 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 samecardinality). After proving that the irrational numbers can be put inone-to-one correspondence with \(\mathbf{R}\), and, surprisingly, thatalso \(\mathbf{R}^{n} \equiv \mathbf{R}\), Cantor conjectured in 1878that any subset of \(\mathbf{R}\) would be either denumerable\((\equiv \mathbf{N})\) or \(\equiv \mathbf{R}\). This is the firstand weakest form of the celebratedContinuum 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 theCantor-Bendixson theorem.

A point set \(P\) isclosed iff its derived set \(P'\subseteq P\), andperfect iff \(P = P'\). TheCantor-Bendixson theorem then states that a closed point set can bedecomposed into two subsets \(R\) and \(S\), such that \(R\) isdenumerable 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 setproperty. Furthermore, Cantor was able to prove that perfect sets havethe power of the continuum (1884). Both results implied that theContinuum Hypothesis is valid for all closed point sets. Many yearslater, in 1916, Pavel Aleksandrov and Felix Hausdorff were able toshow that the broader class of Borel sets have the perfect setproperty too.

His work on points sets led Cantor, in 1882, to conceive of thetransfinite numbers (see Ferreirós 1999:267ff). This was a turning point in his research, for fromthen onward he studiedabstract set theory independently ofmore specific questions having to do with point sets and theirtopology (until the mid-1880s, these questions had been prominent inhis agenda). Subsequently, Cantor focused on the transfinite cardinaland ordinal numbers, and on general order types, independently of thetopological properties of \(\mathbf{R}\).

The transfinite ordinals were introduced as new numbers in animportant mathematico-philosophical paper of 1883,Grundlageneiner allgemeinen Mannigfaltigkeitslehre (notice that Cantorstill uses Riemann’s term Mannigfaltigkeit or‘manifold’ to denote sets). Cantor defined them by meansof two “generating principles”: the first (1) yields thesuccessor \(a+1\) for any given number \(a\), while the second (2)stipulates that there is a number \(b\) which follows immediatelyafter any given sequence of numbers without a last element. Thus,after all the finite numbers comes, by (2), the first transfinitenumber, \(\omega\) (read: omega); and this is followed by\(\omega+1\), \(\omega+2\), …, \(\omega+\omega = \omega \cdot2\), …, \(\omega \cdot n\), \(\omega\cdot n +1\), …,\(\omega^{2}\), \(\omega^{2}+1\), …, \(\omega^{\omega}\),… and so on and on. Whenever a sequence without last elementappears, 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 tomost of his contemporaries, but for Cantor they served two veryimportant functions. To this end, he classified the transfiniteordinals as follows: the “first number class” consisted ofthe finite ordinals, the set \(\mathbf{N}\) of natural numbers; the“second number class” was formed by ω and allnumbers following it (including \(\omega^{\omega}\), and many more)that haveonly a denumerable set of predecessors. Thiscrucial condition was suggested by the problem of proving theCantor-Bendixson theorem (see Ferreirós 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, ifyou write \(\textit{card}(\mathbf{N}) = \aleph_{0}\) (read: alephzero), 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 beproved to be \(\aleph_{2}\). And so on. The first function of thetransfinite ordinals was, thus, to establish a well-defined scale ofincreasing transfinite cardinalities. (The aleph notation used abovewas introduced by Cantor only in 1895.) This made it possible toformulate much more precisely the problem of the continuum;Cantor’s conjecture became the hypothesis that\(\textit{card}(\mathbf{R}) = \aleph_{1}\). Furthermore, relying onthe transfinite ordinals, Cantor was able to prove theCantor-Bendixson theorem, rounding out the results on point sets thathe had been elaborating during these crucial years. TheCantor-Bendixson theorem states: closed sets of \(\mathbf{R}^n\)(generalizable to Polish spaces) have the perfect set property, sothat any closed set \(S\) in \(\mathbf{R}^n\) can be written uniquelyas the disjoint union of a perfect set \(P\) and a countable set\(R\). Moreover, \(P\) is \(S^α\) for α countableordinal.

The study of the transfinite ordinals directed Cantor’sattention towards ordered sets, and in particularwell-orderedsets. A set \(S\) is well-ordered by a relation < iff < isa total order and every subset of \(S\) has a least element in the<-ordering. (The real numbers are not well-ordered in their usualorder: just consider an open interval. Meanwhile, \(\mathbf{N}\) isthe simplest infinite well-ordered set.) Cantor argued that thetransfinite ordinals truly deserve the name ofnumbers,because they express the “type of order” of any possiblewell-ordered set. Notice also that it was easy for Cantor to indicatehow to reorder the natural numbers so as to make them correspond tothe order types \(\omega+1\), \(\omega+2\), …, \(\omega \cdot2\), …, \(\omega \cdot n\), …, \(\omega^2\), …,\(\omega^{\omega}\), … 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 thatthe set \(\mathbf{R}\) of real numbers can indeed be well-ordered.Cantor was so committed to this viewpoint, that he presented thefurther hypothesis thatevery set can be well-ordered as“a fundamental and momentous law of thought”. Some yearslater, Hilbert called attention to both the Continuum Hypothesis andthe well-ordering problem as Problem 1 in his celebrated list of‘Mathematische Probleme’ (1900). Doing so was anintelligent way of emphasizing the importance of set theory for thefuture of mathematics, and the fruitfulness of its new methods andproblems.

In 1895 and 1897, Cantor published his last two articles. They were awell-organized presentation of his results on the transfinite numbers(cardinals and ordinals) and their theory, and also on order types andwell-ordered sets. However, these papers did not advance significantnew ideas. Unfortunately, Cantor had doubts about a third part he hadprepared, which would have discussed very important issues having todo with the problem of well-ordering and the paradoxes (see below).Surprisingly, Cantor also failed to include in the 1895/97 papers atheorem which he had published some years before which is known simplyas Cantor’s Theorem: given any set \(S\), there exists anotherset whose cardinality is greater (this is the power set\(\mathcal{P}(S)\), as we now say—Cantor used instead the set ofall functions of the form \(f\): \(S \rightarrow \{0, 1\}\), which isequivalent). In the same short paper (1892), Cantor presented hisfamous proof that \(\mathbf{R}\) is non-denumerable by the method ofdiagonalisation, a method which he then extended to proveCantor’s Theorem. (A related form of argument had appearedearlier 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 byset theory for the foundations of mathematics. Most important wasDedekind’s contribution (1888) with a deep presentation of thetheory of the natural numbers. He formulated some basic principles ofset (and mapping) theory; gave axioms for the natural number system;proved that mathematical induction is conclusive and recursivedefinitions are flawless; developed the basic theory of arithmetic;introduced the finite cardinals; and proved that his axiom system iscategorical. His system had four axioms. Given a function φdefined on \(S\), a set \(N \subseteq S\), and a distinguished element\(1 \in N\), they are as follows:

Condition (β) is crucial since it ensures minimality for the setof natural numbers, which accounts for the validity of proofs bymathematical induction. \(N = \phi_{o}\{1\}\) is read: \(N\) is thechain of singleton {1} under the function φ, that is, theminimal closure of {1} under the function φ. In general, oneconsiders the chain of a set \(A\) under an arbitrary mapping γ,denoted by \(\gamma_{o}(A)\); in his booklet Dedekind developed aninteresting theory of such chains, which allowed him to prove theCantor-Bernstein theorem. The theory was later generalized by Zermeloand applied by Skolem, Kuratowski, etc.

In the following years, Giuseppe Peano gave a more superficial (butalso more famous) treatment of the natural numbers, employing the newsymbolic language of logic, and Gottlob Frege elaborated his own deepideas, which however fell prey to the paradoxes. An important bookinspired by the set-theoretic style of thinking was Hilbert’sGrundlagen der Geometrie (1899), which took the“mathematics of axioms” one step beyond Dedekind through arich study of geometric systems motivated by questions concerning theindependence of his axioms. Hilbert’s book made clear the newaxiomatic methodology that had been shaping up in connection with thenovel methods of set theory, and he combined it with the axiomatictrends coming from projective geometry.

Nevertheless, as we said before, there was quite a lot of criticism ofset-theoretic, infinitarian methods. As early as 1870, Kronecker hadbegun to voice critical remarks of a constructivist bent that, manyyears later, would be echoed by prominent thinkers like Brouwer orWittgenstein. Kronecker’s critical orientation pointed in theway of renouncing the real number system and classical analysis, infavor of some more stringent form of analysis — twentiethcentury examples of this would be predicative analysis (H. Weylbuilding on basic notions of Poincaré, see Feferman 1988) andintuitionistic analysis (Brouwer). Even Weierstrass had objections (in1874, at least) against the idea of distinguishing sizes of infinity,and that on the face of Cantor’s proofs. Examples abound, and soduring the 1900s many mathematicians expressed doubts about key ideasand methods of set theory. A prototype case is E. Borel, who afterintroducing the ideas of Cantor in France [1898], became increasinglysuspicious 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 Poincaré, Weyl, Skolem,and so on. Among philosophers, the most prominent example isWittgenstein, who condemned set theory for building on the “nonsense”of fictitious symbolism, suggesting “wrong imagery”, andso on.

3. Critical Period

In the late nineteenth century, it was a widespread idea that puremathematics is nothing but an elaborate form of arithmetic. Thus itwas usual to talk about the “arithmetisation” ofmathematics, and how it had brought about the highest standards ofrigor. With Dedekind and Hilbert, this viewpoint led to the idea ofgrounding all of pure mathematics in set theory. The mostdifficult steps in bringing forth this viewpoint had been theestablishment of a theory of the real numbers, and a set-theoreticreduction of the natural numbers. Both problems had been solved by thework of Cantor and Dedekind. But precisely when mathematicians werecelebrating that “full rigor” had been finally attained,serious problems emerged for the foundations of set theory. FirstCantor, 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. Eachtransfinite ordinal is the order type of the set of its predecessors;e.g., ω 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 ofordinals, there corresponds an immediately greater ordinal. Now, the“whole series” of all transfinite ordinals would form awell-ordered set, and to it there would correspond a new ordinalnumber. This is unacceptable, for this ordinal \(o\) would have to begreater than all members of the “whole series”, and inparticular \(o < o\). This is usually called theBurali-Fortiparadox, or paradox of the ordinals (although Burali-Fortihimself failed to formulate it clearly, see Moore & Garciadiego1981).

Although it is conceivable that Cantor might have found that paradoxas early as 1883, immediately after introducing the transfiniteordinals (for arguments in favour of this idea see Purkert &Ilgauds 1987 and Tait 2000), the evidence indicates clearly that itwas not until 1896/97 that he found this paradoxical argument andrealized its implications. By this time, he was also able to employCantor’s Theorem to yield theCantor paradox, orparadox of the alephs: if there existed a “set of all”cardinal numbers (alephs), Cantor’s Theorem applied to it wouldgive a new aleph \(\aleph\), such that \(\aleph < \aleph\). Thegreat set theorist realized perfectly well that these paradoxes were afatal blow to the “logical” approaches to sets favoured byFrege and Dedekind. Cantor emphasized that his views were“in diametrical opposition” to Dedekind’s,and in particular to his “naïve assumption thatallwell-defined collections, or systems, are also‘consistent systems’ ” (see the letter toHilbert, Nov. 15, 1899, in Purkert & Ilgauds 1987: 154). (Contraryto what has often been claimed, Cantor’s ambiguous definition ofset in his paper of 1895 was intended to be “diametricallyopposite” to the logicists’ understanding ofsets—often called “naïve” set theory, whichcould more properly be called the dichotomy conception of sets,following a suggestion of Gödel.)

Cantor thought he could solve the problem of the paradoxes bydistinguishing between “consistent multiplicities” orsets, and “inconsistent multiplicities”. But, in theabsence of explicit criteria for the distinction, this was simply averbal answer to the problem. Being aware of deficiencies in his newideas, Cantor never published a last paper he had been preparing, inwhich he planned to discuss the paradoxes and the problem ofwell-ordering (we know quite well the contents of this unpublishedpaper, as Cantor discussed it in correspondence with Dedekind andHilbert; see the 1899 letters to Dedekind in Cantor 1932, or Ewald1996: vol. 2). Cantor presented an argument that relied on the“Burali-Forti” paradox of the ordinals, and aimed to provethat every set can be well-ordered. This argument was laterrediscovered by the British mathematician P.E.B. Jourdain, but it isopen to criticism because it works with “inconsistentmultiplicities” (Cantor’s term in the above-mentionedletters).

Cantor’s paradoxes convinced Hilbert and Dedekind that therewere 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 Göttingencircle. 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 BertrandRussell, who was led to it by a careful study of Cantor’sTheorem, which conflicted deeply with Russell’s belief in auniversal set. Some time later, in June 1902, he communicated the“contradiction” to Gottlob Frege, who was completing hisown logical foundation of arithmetic, in a well-known letter [vanHeijenoort 1967, 124]. Frege’s reaction made very clear theprofound 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 scientificfoundation, how numbers could be conceived as logical objects”(Frege 1903: 253).

The publication of Volume II of Frege’sGrundgesetze(1903), and above all Russell’s workThe Principles ofMathematics (1903), made the mathematical community fully awareof the existence of the set-theoretic paradoxes, of their impact andimportance. There is evidence that, up to then, even Hilbert andZermelo had not fully appreciated the damage. Notice that theRussell-Zermelo paradox operates with very basicnotions—negation and set membership—concepts that hadwidely been regarded as purely logical. The “set” \(R =\{x: x \not\in x\}\) exists according to the principle ofcomprehension (which allows any open sentence to determine a class),but if so, \(R \in R\)iff \(R \not\in R\). It is a directcontradiction 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. Thedifferent competing viewpoints were widely divergent. Cantor had ametaphysical understanding of set theory and, although he had one ofthe sharpest views of the field, he could not offer a precisefoundation. It was clear to him (as it had been, somewhatmysteriously, to Ernst Schröder in hisVorlesungen überdie Algebra der Logik, 1891) that one has to reject the idea of aUniversal Set, favoured by Frege and Dedekind. Frege and Russell basedtheir approach on the principle of comprehension, which was showncontradictory. Dedekind avoided that principle, but he postulated thatthe Absolute Universe was a set, a “thing” in histechnical sense ofGedankending; and he coupled thatassumption with full acceptance of arbitrary subsets.

This idea of admitting arbitrary subsets had been one of the deepinspirations of both Cantor and Dedekind, but none of them hadthematized it. (Here, their modern understanding of analysis played acrucial but implicit background role, since they worked within theDirichlet-Riemann tradition of “arbitrary” functions.) Asfor the now famous iterative conception there were some elements of it(particularly in Dedekind’s work, with his iterative developmentof the number system, and his views on “systems” and“things”), but it was conspicuously absent from many ofthe relevant authors. Typically, e. g., Cantor did not iterate theprocess of set formation: he tended to consider sets ofhomogeneous elements, elements which were taken to belong“in some conceptual sphere” (either numbers, or points, orfunctions, or even physical particles—but not intermingled). Theiterative conception was first suggested by Kurt Gödel in [1933],in connection with technical work by von Neumann and Zermelo a fewyears earlier; Gödel would insist on the idea in his well-knownpaper on Cantor’s continuum problem. It came onlypostfacto, after very substantial amounts of set theory had beendeveloped and fully systematized.

This variety of conflicting viewpoints contributed much to the overallconfusion, but there was more. In addition to the paradoxes discussedabove (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 dueto Russell, Richard, König, Berry, Grelling, etc., as well as theancient liar paradox due to Epimenides. And the diagnoses and proposedcures for the damage were tremendously varied. Some authors, likeRussell, thought it was essential to find a new logical system thatcould solve all the paradoxes at once. This led him into the ramifiedtype theory that formed the basis ofPrincipia Mathematica (3volumes, Whitehead and Russell 1910–1913), his joint work withAlfred Whitehead. Other authors, like Zermelo, believed that most ofthose paradoxes dissolved as soon as one worked within a restrictedaxiomatic system. They concentrated on the “set-theoretic”paradoxes (as we have done above), and were led to search foraxiomatic systems of set theory.

Even more importantly, the questions left open by Cantor andemphasized by Hilbert in his first problem of 1900 caused heateddebate. At the International Congress of Mathematicians at Heidelberg,1904, Gyula (Julius) König proposed a very detailed proof thatthe cardinality of the continuumcannot be any ofCantor’s alephs. His proof was only flawed because he had reliedon a result previously “proven” by Felix Bernstein, astudent of Cantor and Hilbert. It took some months for Felix Hausdorffto identify the flaw and correct it by properly stating the specialconditions under which Bernstein’s result was valid (seeHausdorff 2001, vol. 1). Once thus corrected, König’stheorem became one of the very few results restricting the possiblesolutions 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 bewell-ordered, using the Axiom of Choice [1904]. During the followingyear, prominent mathematicians in Germany, France, Italy and Englanddiscussed 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 eachset in \(A\). This started a whole era during which the Axiom ofChoice was treated most carefully as a dubious hypothesis (see themonumental study by Moore 1982). And that is ironic, for, among all ofthe usual principles of set theory, the Axiom of Choice is the onlyone that explicitly enforces the existence of some arbitrary subsets.But, important as this idea had been in motivating Cantor andDedekind, and however entangled it is with classical analysis,infinite arbitrary subsets were rejected by many other authors. Amongthe most influential ones in the following period, one ought toemphasize the names of Russell, Hermann Weyl, and of courseBrouwer.

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 manyimportant theorems of analysis (this became gradually clear with workssuch as Sierpiński [1918]). On the other hand, it has ratherunintuitive consequences, such as the Banach-Tarski Paradox, whichsays that the unit ball can be partitioned into finitely-many‘pieces’ (subsets), which can then be rearranged to formtwo unit balls (see Tomkowicz & Wagon [2019]). The objections tothe axiom arise from the fact that it asserts the existence of setsthat cannot be explicitly defined. During the 1920s and 1930s, thereexisted the ritual practice of mentioning it explicitly, whenever atheorem would depend on the axiom. This stopped only afterGödel’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 thefoundations of mathematics, led Zermelo to concentrate on axiomaticset theory. As a result of his incisive analysis, in 1908 he publishedhis axiom system, showing how it blocked the known paradoxes and yetallowed for a masterful development of the theory of cardinals andordinals. This, however, is the topic of the entryZermelo’s axiomatization of set theory; also, on the life and work of Zermelo, see Ebbinghaus 2015.

4. From Zermelo to Gödel

In the period 1900–1930, the rubric “set theory” wasstill understood to include topics in topology and the theory offunctions. Although Cantor, Dedekind, and Zermelo had left that stagebehind to concentrate on pure set theory, for mathematicians at largethis would still take a long time. Thus, at the first InternationalCongress of Mathematicians, 1897, keynote speeches given by Hadamardand Hurwitz defended set theory on the basis of its importance foranalysis. Around 1900, motivated by topics in analysis, important workwas done by three French experts: Borel [1898], Baire [1899] andLebesgue [1902] [1905]. Their work inaugurated the development ofdescriptive set theory by extending Cantor’s studies ondefinable sets of real numbers (in which he had established that theContinuum Hypothesis is valid for closed sets). They introduced thehierarchy of Borel sets, the Baire hierarchy of functions, and theconcept of Lebesgue measure—a crucial concept of modernanalysis.

Descriptive set theory (DST) is the study of certain kinds ofdefinable sets of real numbers, which are obtained from simple kinds(like the open sets and the closed sets) by well-understood operationslike complementation or projection. TheBorel sets were thefirst hierarchy of definable sets, introduced in the 1898 book ofÉmile Borel; they are obtained from the open sets by iteratedapplication of the operations of countable union and complementation.In 1905 Lebesgue studied the Borel sets in an epochal memoir, showingthat their hierarchy has levels for all countable ordinals, andanalyzing the Baire functions as counterparts of the Borel sets. Themain aim of descriptive set theory is to find structural propertiescommon to all such definable sets: for instance, the Borel sets wereshown to have the perfect set property (if uncountable, they have aperfect 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 “regularityproperties” studied in DST are the property of being Lebesguemeasurable, and the so-called property of Baire (to differ from anopen set by a so-called meager set, or set of first category).

Also crucial at the time was the study of theanalytic sets,namely the continuous images of Borel sets, or equivalently, theprojections of Borel sets. The young Russian mathematician MikhailSuslin found a mistake in Lebesgue’s 1905 memoir when herealized that the projection of a Borel set is not Borel in general[Suslin 1917]. However, he was able to establish that the analyticsets, too, possess the perfect set property and thus verify CH. By1923 Nikolai Lusin and Wacław Sierpiński were studying theco-analytic sets, and this was to lead them to a newhierarchy ofprojective sets, which starts with the analyticsets \((\Sigma^{1}_{1})\), 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 ofsets, mainly by Polish mathematicians around Sierpiński and bythe Russian school of Lusin and his students. A crucial resultobtained by Sierpiński was that every \(\Sigma^{1}_{2}\) set isthe union of \(\aleph_{1}\) Borel sets (the same holds for\(\Sigma^{1}_{1}\) sets), but this kind of traditional research on thetopic would stagnate after around 1940 (see Kanamori [1995]).

Soon Lusin, Sierpiński and their colleagues were finding extremedifficulties in their work. Lusin was so much in despair that, in apaper of 1925, he came to the “totally unexpected”conclusion that “one does not know and one will neverknow” whether the projective sets have the desired regularityproperties (quoted in Kanamori 1995: 250). Such comments are highlyinteresting in the light of later developments, which have led tohypotheses that solve all the relevant questions (ProjectiveDeterminacy, in particular). They underscore the difficultmethodological and philosophical issues raised by these more recenthypotheses, namely the problem concerning the kind of evidence thatbacks them.

Lusin summarized the state of the art in his 1930 bookLeçons 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 forthe Baire space \( ^{\omega}\)\(\omega\) of infinite sequences ofnatural numbers, which in effect had been introduced by RenéBaire in a paper published in 1909. Baire space is endowed with acertain topology that makes it homeomorphic to the set of theirrational numbers, and it is regarded by experts to be “perhapsthe most fundamental object of study of set theory” next to theset of natural numbers [Moschovakis 1994, 135].

This stream of work on DST must be counted among the most importantcontributions made by set theory to analysis and topology. But whathad begun as an attempt to prove the Continuum Hypothesis could notreach this goal. Soon it was shown using the Axiom of Choice thatthere are non-Lebesgue measurable sets of reals (Vitali 1905), andalso uncountable sets of reals with no perfect subset (Bernstein1908). Such results made clear the impossibility of reaching the goalof CH by concentrating on definable and “well-behaved”sets of reals.

Also, with Gödel’s work around 1940 (and also with forcingin the 1960s) it became clear why the research of the 1920s and 30shad stagnated: the fundamental new independence results showed thatthe theorems established by Suslin (perfect set property for analyticsets), Sierpinski (\(\Sigma^{1}_{2}\) sets as unions of \(\aleph_{1}\)Borel sets) and a few others were the best possible results on thebasis of axiom system ZFC. This is important philosophically: alreadyan exploration of the world of sets definable from the open (orclosed) sets by complement, countable union, and projection hadsufficed to reach the limits of the ZFC system. Hence the need for newaxioms, that Gödel emphasized after World War II [Gödel1947].

Let us now turn to Cantor’s other main legacy, the study oftransfinite numbers. By 1908 Hausdorff was working on uncountableorder types and introduced the Generalized Continuum Hypothesis\((2^{\aleph_{a}} = \aleph_{a+1})\). He was also the first to considerthe possibility of an “exorbitant” cardinal, namely aweakly 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 suchnumbers). Few years later, in the early 1910s, Paul Mahlo was studyinghierarchies of such large cardinals in work that pioneered what was tobecome a central area of set theory; he obtained a succession ofinaccessible cardinals by employing a certain operation that involvesthe notion of a stationary subset; they are called Mahlo cardinals.But the study of large cardinals developed slowly. Meanwhile,Hausdorff’s textbookGrundzüge der Mengenlehre(1914) introduced two generations of mathematicians into set theoryand general topology.

The next crucial steps into the “very high” infinite weredone in 1930. The notion of strongly inaccessible cardinals was thenisolated by Sierpiński & Tarski, and by Zermelo [1930]. Astrong inaccessible is a regular cardinal \(\alpha\) such that \(2^x\)is less than \(\alpha\) whenever \(x < \alpha\). While weakinaccessibles merely involve closure under the successor operation,strong inaccessibles involve a much stronger notion of closure underthe powerset operation. That same year, in a path-breaking paper onmodels of ZFC, Zermelo [1930] established a link between theuncountable (strongly) inaccessible cardinals and certain“natural” models of ZFC (in which work he assumed that thepowerset operation is, so to say, fully determinate).

In that same year, Stanislaw Ulam was led by considerations coming outof analysis (measure theory) to a concept that was to become central:measurable cardinals. It turned out that such cardinals, defined by ameasure-theoretic property, had to be (strongly) inaccessible. Indeed,many years later it would be established (by Hanf, working uponTarski’s earlier work) that the first inaccessible cardinal isnot measurable, showing that these new cardinals were even more“exorbitant”. As one can see, the Polish school led bySierpiński had a very central role in the development of settheory between the Wars. Measurable cardinals came to specialprominence in the late 1960s when it became clear that the existenceof a measurable cardinal contradicts Gödel’s axiom ofconstructibility (\(V = L\) in the class notation). This againvindicated Gödel’s convictions, expressed in what issometimes called “Gödel’s program” for newaxioms.

Set-theoretic mathematics continued its development into the powerfulaxiomatic and structural approach that was to dominate much of the20^th century. To give just a couple of examples,Hilbert’s early axiomatic work (e.g., in his arch-famousFoundations of Geometry) was deeply set-theoretic; ErnstSteinitz published in 1910 his research on abstract field theory,making essential use of the Axiom of Choice; and around the same timethe study of function spaces began with work by Hilbert, MauriceFréchet, and others. During the 1920s and 30s, the firstspecialized mathematics journal,Fundamenta Mathematicae, wasdevoted to set theory as then understood (centrally including topologyand function theory). In those decades structural algebra came of age,abstract topology was gradually becoming an independent branch ofstudy, and the study of set theory initiated its metatheoreticturn.

Ever since, “set theory” has generally been identifiedwith the branch of mathematical logic that studies transfinite sets,originating in Cantor’s result that \(\mathbf{R}\) has a greatercardinality than \(\mathbf{N}\). But, as the foregoing discussionshows, set theory was both effect and cause of the rise of modernmathematics: the traces of this origin are indelibly stamped on itsaxiomatic structure.

Bibliography

Cited Works

• Alexandroff, Pavel, 1916, “Sur la puissance des ensemblesmesurables B”,Comptes Rendus Acad. Sci. Paris, 162:323–325.
• Asghari, Amir, 2019, “Equivalence: an attempt at a historyof the idea”,Synthese, 196: 4657–4677.
• Baire, René, 1899, “Sur les fonctions de variablesreelles”,Annali di Matematica Pura ed Applicata, SerieIIIa, vol. 3, pp. 1–122.
• –––, 1909, “Sur la représentationdes fonctions discontinues”,Acta Mathematica, 32:97–176.
• Bernstein, Felix, 1908, “Zur Theorie der trigonometrischenReihen”,Sitzungsberichte der Königlich SachsischenGesellschaft der Wissenschaften zu Leipzig, Math.-Phys. Klasse,60: 325–338.
• du Bois–Reymond, Paul, 1875, “Ueber asymptotischeWerthe, infinitäre Approximationen und infinitäreAuflösung von Gleichungen”,Mathematische Annalen,8: 363–414.
• Bolzano, Bernard, 1851,Paradoxien des Unendlichen,Leipzig, Reclam; English translation London, Routledge, 1920.
• Borel, Émile, 1898,Leçons sur la théoriedes fonctions, Paris, Gauthier-Villars. 4^th edn 1950with numerous additions.
• Cantor, Georg, 1872, “Über die Ausdehnung eines Satzesaus der Theorie der trigonometrischen Reihen”,MathematischeAnnalen, 5: 123–132. In Cantor 1932: 92–102.
• –––, 1883,Grundlagen einer allgemeinenMannigfaltigkeitslehre, Leipzig: B.G. Teubner. In Cantor 1932:165–208. English translation in Ewald 1996: vol. 2.
• –––, 1884, “Über unendliche, linearePunktmannichfaltigkeiten, 6”,Mathematische Annalen,23: 453–88. Reprinted in Cantor 1932: 210–244.
• –––, 1892, “Über eine elementareFrage der Mannigfaltigkeitslehre”,Jahresbericht derDeutschen Mathematiker Vereinigung, 1: 75–78. Reprinted inCantor 1932: 278–280. English translation in Ewald 1996:vol.2.
• –––, 1895/97, “Beiträge zurBegründung der transfiniten Mengenlehre”, in Cantor 1932:282–351. English translation in Cantor,Contributions to thefounding of the theory of transfinite numbers, New York: Dover,1955.
• –––, 1932,Gesammelte Abhandlungenmathematischen und philosophischen Inhalts, E. Zermelo (ed.),Berlin: Springer. Reprint Hildesheim: Olms, 1966.
• Dauben, Joseph, 1979,Georg Cantor. His Mathematics andPhilosophy of the Infinite, Cambridge, MA: Harvard UniversityPress.
• Dedekind, Richard, 1871, “Über die Komposition derbinären quadratischen Formen”, Supplement X to G.L.Dirichlet & R. Dedekind,Vorlesungen überZahlentheorie, Braunschweig: Vieweg. [Later editions asSupplement XI, of which the fourth is reprinted in New York: Chelsea,1968.] Partial reprint in Dedekind 1930/32: vol.3, 223–261.
• –––, 1872,Stetigkeit und irrationaleZahlen, Braunschweig: Vieweg. In Dedekind 1930/32: vol.3,315–334. English translation in Ewald 1996: vol. 2.
• –––, 1876/77, “Sur la théorie desnombres entiers algébriques”,Bulletin des Sciencesmathématiques et astronomiques, 1^st series, XI(1876): 278–293; 2^nd series, I (1877): 17–41,69–92, 144–164, 207–248. Separate edition, Paris:Gauthier-Villars, 1977. English translation by J. Stillwell:Theory of algebraic integers, Cambridge: Cambridge UniversityPress, 2004.
• –––, 1888,Was sind und was sollen dieZahlen?, Braunschweig: Vieweg. In Dedekind 1930/32: vol. 3.English in Ewald 1996: vol. 2.
• –––, 1930/32.Gesammelte mathematischeWerke, R. Fricke, E. Noether & Ö. Ore (eds.),Braunschweig: Vieweg, 3 vols. Reprint New York: Chelsea, 1969.
• Dedekind, R. & Heinrich Weber, 1882, “Theorie deralgebraischen Funktionen einer Veränderlichen”,Journalfür reine und angew. Mathematik, 92: 181–290;reprinted in Dedekind 1930/32 (Volume 1), pp. 238–350; Englishtranslation by John Stillwell,Theory of Algebraic Functions ofOne Variable, Providence: American Mathematical Society andLondon Mathematical Society, 2012.
• Deiser, O., 2020,Der Briefwechsel zwischen Cantor undDedekind (online), see theAnhänge
• Drobisch, M. W., 1851,Neue Darstellung der Logik nach ihreneinfachsten Verhältnissen. Nebst einem logisch-mathematischenAnhange, second edition, Leipzig: Voss.
• Ebbinghaus, H. D., 2015,Ernst Zermelo: An approach to hislife and work, second edition, Berlin: Springer Verlag.
• Ewald, William B., 1996,From Kant to Hilbert: A source bookin the foundations of mathematics, 2 volumes, Oxford: OxfordUniversity Press.
• Feferman, Solomon, 1988, “Weyl vindicated: Das Kontinuum 70years later”, reprinted inIn the Light of Logic,Oxford: Oxford University Press, 1998, chap. 13.
• Ferreirós, José, 1993, “On the Relationsbetween Georg Cantor and Richard Dedekind”,HistoriaMathematica, 20. (1993), 343–363.
• –––, 1995, “‘What Fermented in Mefor Years’: Cantor’s Discovery of TransfiniteNumbers”,Historia Mathematica, 22: 33–42.
• –––, 1999,Labyrinth of Thought. A historyof set theory and its role in modern mathematics, Basel:Birkhäuser.
• –––, 2009, “Hilbert, logicism, andmathematical existence”,Synthese, 170,33–70.
• Frege, Gottlob, 1903,Grundgesetze der Arithmetik, vol.2, Jena: Pohle. Reprint Hildesheim: Olms, 1966.
• Gödel, Kurt, 1933, “The present situation in thefoundations of mathematics”, in S. Feferman et al. (eds),Collected Works, Vol. 3, Oxford University Press, pp.45–53.
• –––, 1947, “What is Cantor’scontinuum problem?”,American Mathematical Monthly, 54.Reprinted in S. Feferman et al. (eds),Collected Works, Vol.2, Oxford University Press, pp. 176–187.
• Hallett, Michael, 1984,Cantorian Set Theory and Limitation ofSize, Oxford: Clarendon.
• Hausdorff, Felix, 1914,Grundzüge der Mengenlehre,Leipzig: Viet. Reprinted New York: AMS Chelsea Publishing, 1949.Reprinted as Volume II of Hausdorff 2001–. The third edition(1937) was translated into English, 1957,Set theory, NewYork: AMS Chelsea Publishing.online scan of Hausdorff 1914.
• –––, 1916, “Die Mächtigkeit derBorelschen Mengen”,Mathematische Annalen, 77(3):430–437. In Hausdorff [2001–], vol. 3. doi:10.1007/BF01475871
• –––, 2001–,Gesammelte Werke, 9volumes, E. Brieskorn, W. Purkert, U. Felgner, E. Scholz et al.(eds.), Berlin: Springer.
• van Heijenoort, Jean, 1967,From Frege to Gödel: A sourcebook in mathematical logic, 1879–1931, Cambridge, MA:Harvard University Press. Reprint as paperback, 2000.
• Kanamori, Akihiro, 1995, “The emergence of descriptive settheory”,Synthese, 251: 241–262. .
• –––, 1996, “The mathematical developmentof set theory from Cantor to Cohen”,Bulletin of SymbolicLogic, 2: 1–71.
• Lavine, Shaughan, 1994,Understanding the Infinite,Cambridge, MA: Harvard University Press.
• Lebesgue, Henri, 1902, “Intégrale, longueur,aire”,Annali di Matematica Pura ed Applicata, 7 (1):231–359.
• –––, 1905, “Sur les fonctionsrepreséntables analytiquement”,Journal deMathématiques, (6e serie), 1: 139–216.
• Lusin, Nikolai, 1925, “Sur les ensembles projectifs de M.Lebesgue”,Comptes Rendus Acad. Scie. Paris, 180:1572–74.
• –––, 1930,Leçons sur les EnsemblesAnalytiques et leurs Applications, with a preface by Lebesgue anda note by Sierpinski, Paris: Gauthier-Villars.
• Mancosu, Paolo, 2009, “Measuring the Size of InfiniteCollections of Natural Numbers: Was Cantor’s Theory of InfiniteNumber Inevitable?”,The Review of Symbolic Logic,2(04): 612 – 646.
• Moore, Gregory H., 1982,Zermelo’s Axiom of Choice. ItsOrigins, Development and Influence, Berlin: Springer.
• Moore, Gregory H. & A. Garciadiego, 1981,“Burali-Forti’s Paradox: A reappraisal of itsorigins”,Historia Mathematica, 8: 319–50.
• Moschovakis, Yiannis N., 1994,Set Theory Notes, NewYork: Springer.
• Müller-Stach, S., 2024,Richard Dedekind. What Are andWhat Should the Numbers Be?. Springer Verlag.
• Peckhaus, Volker & R. Kahle, 2002, “Hilbert’sParadox”,Historia Mathematica, 29 (2):157–175.
• Purkert, Walter & H.J. Ilgauds, 1987,Georg Cantor1845–1918, Basel: Birkhäuser.
• Rang, Bernhard & W. Thomas, 1981, “Zermelo’sDiscovery of the ‘Russell Paradox’”,HistoriaMathematica, 8: 15–22.
• Riemann, Bernhard, 1854/1868a, “Über die Hypothesen,welche der Geometrie zu Grunde liegen” (Habilitationsvotrag),Abhandlungen der Königlichen Gesellschaft der Wissenschaftenzu Göttingen, 13 (1868): 133–152. In Riemann 1892:272–287. English translation by Clifford, reprinted in Ewald1996: vol. 2.
• –––, 1854/1868b, “Über dieDarstellbarkeit einer Function durch eine trigonometrischeReihe”, (Habilitationsschrift),Abhandlungen derKöniglichen Gesellschaft der Wissenschaften zuGöttingen, 13 (1868): 87–132. In Riemann 1892:227–265.
• –––, 1892,Gesammelte mathematische Werkeund wissenschaftlicher Nachlass, H. Weber and R. Dedekind (eds.),Leipzig, Teubner. Reprinted (together with theNachträge), M. Noether and W. Wirtinger (eds.), NewYork: Dover, 1953.
• Russell, Bertrand, 1903,The Principles of Mathematics,Cambridge, University Press. Reprint of the 2^nd edn.(1937): London: Allen & Unwin, 1948.
• Sierpiński, Waclav, 1918, “L’axiome de M. Zermeloet son rôle dans la théorie des ensembles etl’analyse”,Bulletin de l’Académie desSciences de Cracovie (Cl. Sci. Math. A), 99–152;reprinted in Sierpiński,Oeuvres choisies, S. Hartman,et al. (eds.), Volume 2, Warszawa: Editions scientifiques de Pologne,1974.
• Sierpiński, Waclav & Alfred Tarski, 1930, “Sur unepropriété caractéristique des nombresinaccessibles”,Fundamenta Mathematicae, 15:292–300.
• Steinitz, Ernst, 1910, “Algebraische Theorie derKörper”,Journal für die reine und angewandteMathematik, 137: 167–309.
• Suslin, Mikhail Ya., 1917, “Sur une définition desensembles measurables B sans nombres transfinis”,ComptesRendues Acad. Sci. Paris, 164: 88–91.
• Tomkowicz, G., and Wagon, S., 2019,The Banach-TarskiParadox, second edition, Cambridge: Cambridge UniversityPress.
• Vitali, G., 1905,Sul problema della misura dei gruppi dipunti di una retta, Bologna: Gamberini e Parmeggiani.
• Whitehead, Alfred N. & Bertrand Russell, 1910–1913,Principia Mathematica, 3 volumes, Cambridge: CambridgeUniversity Press.
• Tait, William W., 2000, “Cantor’sGrundlagenand the Paradoxes of Set Theory”,Between Logic andIntuition: Essays in Honor of Charles Parsons, G. Sher and R.Tieszen (eds), Cambridge: Cambridge University Press, pp.269–290. Reprinted in hisThe Provenance of PureReason, Oxford: Oxford University Press, 2005, pp. 252–275.
• Wang, Hao, 1974, “The concept of set”, inFromMathematics to Philosophy, London, Routledge; reprinted in P.Benacerraf & H. Putnam,Philosophy of Mathematics: selectedreadings, Cambridge Univ. Press, 1983, 530–570.
• Zermelo, Ernst, 1904, “Beweis, dass jede Menge wohlgeordnetwerden kann”,Mathematische Annalen, 59: 514–516;in Zermelo [2010], vol. 1, 80–119. English translation in vanHeijenoort 1967 (“Proof that every set can bewell-ordered”).
• –––, 1908, “Untersuchungen über dieGrundlagen der Mengenlehre”,Mathematische Annalen, 65:261–281; ; in Zermelo [2010], vol. 1, 160–229. Englishtranslation in van Heijenoort 1967 (“Investigations in thefoundations of set theory I”).
• –––, 2010–2011,Collected Works /Gesammelte Werke, Vol. I and II, H.-D. Ebbinghaus et al. (eds.),Springer: Berlín,

Further Reading

• Cavaillès, Jean, 1962,Philosophiemathématique, Paris: Hermann.
• Ebbinghaus, Heinz-Dieter, 2007,Ernst Zermelo: An approach tohis life an work, New York: Springer.
• Fraenkel, Abraham, 1928,Einleitung in die Mengenlehre,3^rd edn. Berlin: Springer.
• Grattan-Guinness, Ivor (ed.), 1980,From the Calculus to SetTheory, 1630–1910, London: Duckworth.
• Kanamori, Akihiro, 2004, “Zermelo and set theory”,Bulletin of Symbolic Logic, 10(4): 487–553.
• –––, 2007, “Gödel and settheory”,Bulletin of Symbolic Logic, 13 (2):153–188.
• –––, 2008, “Cohen and set theory”,Bulletin of Symbolic Logic, 14(3): 351–378.
• –––, 2009, “Bernays and set theory”,Bulletin of Symbolic Logic, 15(1): 43–60.
• Maddy, Penelope, 1988, “Believing the axioms”,Journal of Symbolic Logic, 53(2): 481–511; 53(3):736–764.
• Wagon, Stan, 1993,The Banach-Tarski Paradox, Cambridge:Cambridge University Press.

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Other Internet Resources

• A History of Set Theory, by J.J. O’Connor and E.F. Robertson, in The MacTutor History ofMathematics archive. Note that their reconstruction conflicts at somepoints with the one provided here.
• Godel’s Program (PowerPoint), an interesting talk by John R. Steel (Mathematics,U.C./Berkeley).
• A Home Page for the Axiom of Choice, maintained by Eric Schechter (Mathematics, VanderbiltUniversity).

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