The objections to Cantor's work were occasionally fierce:Leopold Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth".[9] Kronecker objected to Cantor's proofs that the algebraic numbers are countable, and that the transcendental numbers are uncountable, results now included in a standard mathematics curriculum. Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong".[10] Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries,[11] though some have explained these episodes as probable manifestations of abipolar disorder.[12]
Georg Cantor, born in 1845 inSaint Petersburg, Russian Empire, was brought up in that city until the age of eleven. The oldest of six children, he was regarded as an outstanding violinist. His father was a Danish trader and his mother an Austro-Hungarian artist (seeCantor's Ancestry).[16] His maternal grandfather Franz Böhm (1788–1846; the violinistJoseph Böhm's brother) was a well-known musician and a soloist in a Russian imperial orchestra.[17]
Cantor's father had been a member of theSaint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first toWiesbaden, then toFrankfurt, seeking milder winters than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule inDarmstadt; his exceptional skills in mathematics,trigonometry in particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now theTechnische Universität Darmstadt.[18][19] In 1862 Cantor entered theSwiss Federal Polytechnic in Zurich. After receiving a substantial inheritance upon his father's death in June 1863,[20] Cantor transferred to theUniversity of Berlin, attending lectures byLeopold Kronecker,Karl Weierstrass andErnst Kummer. He spent the summer of 1866 at theUniversity of Göttingen, then and later a center for mathematical research. Cantor was a good student, and he received his doctoral degree in 1867.[20][21]
Cantor submitted hisdissertation onnumber theory at the University of Berlin in 1867. After teaching briefly in a Berlin girls' school, he took up a position at theUniversity of Halle, where he spent his entire career. He was awarded the requisitehabilitation for his thesis, also on number theory, which he presented in 1869 upon his appointment at Halle.[21][22]
In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite his modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions withRichard Dedekind, whom he had met at Interlaken in Switzerlandtwo years earlier while on holiday.[23]
Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879.[21][20] To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired achair at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible.[24] Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,[25] perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.[26] Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, disagreed fundamentally with the thrust of Cantor's work ever since he had intentionally delayed the publication of Cantor's first major publication in 1874.[21] Kronecker, now seen as one of the founders of theconstructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Whenever Cantor applied for a post in Berlin, he was declined, and the process usually involved Kronecker,[21] so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.[27]
In 1881, Cantor's Halle colleagueEduard Heine died. Halle accepted Cantor's suggestion that Heine's vacant chair be offered to Dedekind,Heinrich M. Weber andFranz Mertens, in that order, but each declined the chair after being offered it. Friedrich Wangerin was eventually appointed, but he was never close to Cantor.[28]
In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle.[29] Cantor also began another important correspondence, withGösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journalActa Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted toActa.[30] He asked Cantor to withdraw the paper fromActa while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but then curtailed his relationship and correspondence with Mittag-Leffler, writing to a third party, "Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand! ... But of course I never want to know anything again aboutActa Mathematica."[31]
Cantor suffered his first known bout of depression in May 1884.[20][32] Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 mentioned Kronecker. A passage from one of these letters is revealing of the damage to Cantor's self-confidence:
... I don't know when I shall return to the continuation of my scientific work. At the moment I can do absolutely nothing with it, and limit myself to the most necessary duty of my lectures; how much happier I would be to be scientifically active, if only I had the necessary mental freshness.[33]
Cantor recovered soon thereafter, and subsequently made further important contributions, including hisdiagonal argument andtheorem. However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on 29 December 1891.[21] He eventually sought, and achieved, a reconciliation with Kronecker. Nevertheless, the philosophical disagreements and difficulties dividing them persisted.
In 1889, Cantor was instrumental in founding theGerman Mathematical Society,[21] and he chaired its first meeting in Halle in 1891, where he first introduced his diagonal argument; his reputation was strong enough, despite Kronecker's opposition to his work, to ensure he was elected as the first president of this society. Setting aside the animosity Kronecker had displayed towards him, Cantor invited him to address the meeting, but Kronecker was unable to do so because his wife was dying from injuries sustained in a skiing accident at the time. Georg Cantor was also instrumental in the establishment of the firstInternational Congress of Mathematicians, which took place in Zürich, Switzerland, in 1897.[21]
After Cantor's 1884 hospitalization there is no record that he was in anysanatorium again until 1899.[32] Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on 16 December (Cantor was delivering a lecture on his views onBaconian theory andWilliam Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics.[35] Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented byJulius König at the ThirdInternational Congress of Mathematicians. The paper attempted to prove that the basic tenets oftransfinite set theory were false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated.[36] AlthoughErnst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God.[13] Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined to various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years.[37] He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox,Cantor's paradox, andRussell's paradox) to a meeting of theDeutsche Mathematiker-Vereinigung in 1903, and attending the International Congress of Mathematicians atHeidelberg in 1904.
In 1911, Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of theUniversity of St. Andrews in Scotland. Cantor attended, hoping to meetBertrand Russell, whose newly publishedPrincipia Mathematica repeatedly cited Cantor's work, but the encounter did not come about. The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.
Cantor retired in 1913, and lived in poverty and suffered frommalnourishment duringWorld War I.[38] The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on 6 January 1918, in the sanatorium where he had spent the last year of his life.[20]
Cantor's work between 1874 and 1884 is the origin ofset theory.[39] Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginning of mathematics, dating back to the ideas ofAristotle. No one had realized that set theory had any nontrivial content. Before Cantor, there were only finite sets (which are easy to understand) and "the infinite" (which was considered a topic for philosophical, rather than mathematical, discussion). By proving that there are (infinitely) many possible sizes for infinite sets, Cantor established that set theory was not trivial, and it needed to be studied.Set theory has come to play the role of afoundational theory in modern mathematics, in the sense that it interprets propositions about mathematical objects (for example, numbers and functions) from all the traditional areas of mathematics (such asalgebra,analysis, andtopology) in a single theory, and provides a standard set of axioms to prove or disprove them. The basic concepts of set theory are now used throughout mathematics.[40]
In one of his earliest papers,[41] Cantor proved that the set ofreal numbers is "more numerous" than the set ofnatural numbers; this showed, for the first time, that there exist infinite sets of differentsizes. He was also the first to appreciate the importance ofone-to-one correspondences (hereinafter denoted "1-to-1 correspondence") in set theory. He used this concept to definefinite andinfinite sets, subdividing the latter intodenumerable (or countably infinite) sets andnondenumerable sets (uncountably infinite sets).[42]
Cantor introduced fundamental constructions in set theory, such as thepower set of a setA, which is the set of all possiblesubsets ofA. He later proved that the size of the power set ofA is strictly larger than the size ofA, even whenA is an infinite set; this result soon became known asCantor's theorem. Cantor developed an entire theory andarithmetic of infinite sets, calledcardinals andordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter (ℵ,aleph) with a natural number subscript; for the ordinals he employed the Greek letter (ω,omega). This notation is still in use today.
TheContinuum hypothesis, introduced by Cantor, was presented byDavid Hilbert as the first of histwenty-three open problems in his address at the 1900International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium.[15] The US philosopherCharles Sanders Peirce praised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zürich in 1897,Adolf Hurwitz andJacques Hadamard also both expressed their admiration. At that Congress, Cantor renewed his friendship and correspondence with Dedekind. From 1905, Cantor corresponded with his British admirer and translatorPhilip Jourdain on the history ofset theory and on Cantor's religious ideas. This was later published, as were several of his expository works.
Cantor's first ten papers were onnumber theory, his thesis topic. At the suggestion ofEduard Heine, the Professor at Halle, Cantor turned toanalysis. Heine proposed that Cantor solvean open problem that had eludedPeter Gustav Lejeune Dirichlet,Rudolf Lipschitz,Bernhard Riemann, and Heine himself: the uniqueness of the representation of afunction bytrigonometric series. Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indicesn in thenthderived setSn of a setS of zeros of a trigonometric series. Given a trigonometric series f(x) withS as its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that hadS1 as its set of zeros, whereS1 is the set oflimit points ofS. IfSk+1 is the set of limit points ofSk, then he could construct a trigonometric series whose zeros areSk+1. Because the setsSk were closed, they contained their limit points, and the intersection of the infinite decreasing sequence of setsS,S1,S2,S3,... formed a limit set, which we would now callSω, and then he noticed thatSω would also have to have a set of limit pointsSω+1, and so on. He had examples that went on forever, and so here was a naturally occurring infinite sequence of infinite numbersω,ω + 1,ω + 2, ...[44]
Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper definingirrational numbers asconvergent sequences ofrational numbers. Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers byDedekind cuts. While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories ofinfinitesimals of his contemporariesOtto Stolz andPaul du Bois-Reymond, describing them as both "an abomination" and "acholerabacillus of mathematics".[45] Cantor also published an erroneous "proof" of the inconsistency ofinfinitesimals.[46]
The beginning of set theory as a branch of mathematics is often marked by the publication ofCantor's 1874 paper,[39] "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").[48] This paper was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to beequinumerous (that is, of "the same size" or having the same number of elements).[49] Cantor proved that the collection of real numbers and the collection of positiveintegers are not equinumerous. In other words, the real numbers are notcountable. His proof differs from thediagonal argument that he gave in 1891.[50] Cantor's article also contains a new method of constructingtranscendental numbers. Transcendental numbers were first constructed byJoseph Liouville in 1844.[51]
Cantor established these results using two constructions. His first construction shows how to write the realalgebraic numbers[52] as asequencea1,a2,a3, .... In other words, the real algebraic numbers are countable. Cantor starts his second construction with any sequence of real numbers. Using this sequence, he constructsnested intervals whoseintersection contains a real number not in the sequence. Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence – that is, the real numbers are not countable. By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number. Cantor points out that his constructions prove more – namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.[53] Cantor's next article contains a construction that proves the set of transcendental numbers has the same "power" (see below) as the set of real numbers.[54]
Between 1879 and 1884, Cantor published a series of six articles inMathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Leopold Kronecker, who admitted mathematical concepts only if they could be constructed in afinite number of steps from the natural numbers, which he took as intuitively given. For Kronecker, Cantor's hierarchy of infinities was inadmissible, since accepting the concept ofactual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole.[55] Cantor also introduced theCantor set during this period.
The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre" ("Foundations of a General Theory of Aggregates"), published in 1883,[56] was the most important of the six and was also published as a separatemonograph. It contained Cantor's reply to his critics and showed how thetransfinite numbers were a systematic extension of the natural numbers. It begins by definingwell-ordered sets.Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of thecardinal and ordinal numbers. In 1885, Cantor extended his theory of order types so that the ordinal numbers simply became a special case of order types.
In 1891, he published a paper containing his elegant "diagonal argument" for the existence of an uncountable set. He applied the same idea to proveCantor's theorem: thecardinality of the power set of a setA is strictly larger than the cardinality ofA. This established the richness of the hierarchy of infinite sets, and of thecardinal andordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of theHalting problem and the proof ofGödel's first incompleteness theorem. Cantor wrote on theGoldbach conjecture in 1894.
Passage of Georg Cantor's article with his set definition
In 1895 and 1897, Cantor published a two-part paper inMathematische Annalen underFelix Klein's editorship; these were his last significant papers on set theory.[57] The first paper begins by defining set,subset, etc., in ways that would be largely acceptable now. The cardinal and ordinal arithmetic are reviewed. Cantor wanted the second paper to include a proof of the continuum hypothesis, but had to settle for expositing his theory ofwell-ordered sets and ordinal numbers. Cantor attempts to prove that ifA andB are sets withAequivalent to a subset ofB andB equivalent to a subset ofA, thenA andB are equivalent.Ernst Schröder had stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed.Felix Bernstein supplied a correct proof in his 1898 PhD thesis; hence the nameCantor–Bernstein–Schröder theorem.
Cantor's 1874Crelle paper was the first to invoke the notion of a1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of theunit square and the points of a unitline segment. In an 1877 letter to Richard Dedekind, Cantor proved a farstronger result: for any positive integern, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in ann-dimensional space. About this discovery Cantor wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!")[58] The result that he found so astonishing has implications for geometry and the notion ofdimension.
In 1878, Cantor submitted another paper to Crelle's Journal, in which he defined precisely the concept of a 1-to-1 correspondence and introduced the notion of "power" (a term he took fromJakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor definedcountable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with thenatural numbers, and proved that the rational numbers are denumerable. He also proved thatn-dimensionalEuclidean spaceRn has the same power as thereal numbersR, as does a countably infiniteproduct of copies ofR. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking aboutdimension, stressing that hismapping between theunit interval and the unit square was not acontinuous one.
This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so andKarl Weierstrass supported its publication.[59] Nevertheless, Cantor never again submitted anything to Crelle.
Cantor was the first to formulate what later came to be known as thecontinuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals isexactly aleph-one, rather than justat least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years toprove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.[11]
The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result byKurt Gödel and a 1963 one byPaul Cohen together imply that the continuum hypothesis can be neither proved nor disproved using standardZermelo–Fraenkel set theory plus theaxiom of choice (the combination referred to as "ZFC").[60]
Absolute infinite, well-ordering theorem, and paradoxes
In 1883, Cantor divided the infinite into the transfinite and theabsolute.[61]
The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, the ordinals form an absolutely infinite sequence that cannot be increased in magnitude because there are no larger ordinals to add to it.[62] In 1883, Cantor also introduced thewell-ordering principle "every set can be well-ordered" and stated that it is a "law of thought".[63]
Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is analeph.[64] First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem.[65] In 1932, Zermelo criticized the construction in Cantor's proof.[66]
Cantor avoidedparadoxes by recognizing that there are two types of multiplicities. In his set theory, when it is assumed that the ordinals form a set, the resulting contradiction implies only that the ordinals form an inconsistent multiplicity. In contrast,Bertrand Russell treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory isinconsistent. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: theBurali-Forti paradox (which was just mentioned),Cantor's paradox, andRussell's paradox.[67] Russell named paradoxes afterCesare Burali-Forti and Cantor even though neither of them believed that they had found paradoxes.[68]
In 1908, Zermelo publishedhis axiom system for set theory. He had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of thewell-ordering theorem.[69] Zermelo had proved this theorem in 1904 using theaxiom of choice, but his proof was criticized for a variety of reasons.[70] His response to the criticism included his axiom system and a new proof of the well-ordering theorem. His axioms support this new proof, and they eliminate the paradoxes by restricting the formation of sets.[71]
In 1923,John von Neumann developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently. Von Neumann stated that aclass is too big to be a set if it can be put into one-to-one correspondence with the class of all sets. He defined a set as a class that is a member of some class and stated the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets. This axiom implies that these big classes are not sets, which eliminates the paradoxes since they cannot be members of any class.[72] Von Neumann also used his axiom to prove the well-ordering theorem: Like Cantor, he assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem.[73] In 1930, Zermelo definedmodels of set theory that satisfy von Neumann's axiom.[74]
Philosophy, religion, literature and Cantor's mathematics
The concept of the existence of anactual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving theorthodoxy of the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's.[75] He directly addressed this intersection between these disciplines in the introduction to hisGrundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one.[76] To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications – he identified theabsolute infinite with God,[77] and he considered his work on transfinite numbers to have been directly communicated to him by God, who had chosen Cantor to reveal them to the world.[5] He was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science.[78]Joseph Dauben has traced the effect Cantor's Christian convictions had on the development of transfinite set theory.[79][80]
Debate among mathematicians grew out of opposing views in thephilosophy of mathematics regarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence.[81] Mathematicians from three major schools of thought (constructivism and its two offshoots,intuitionism andfinitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea thatnonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead thatconstructive proofs are required. Intuitionism also rejects the idea that actual infinity is an expression of any sort of reality, but arrive at the decision via a different route than constructivism. Firstly, Cantor's argument rests on logic to prove the existence of transfinite numbers as an actual mathematical entity, whereas intuitionists hold that mathematical entities cannot be reduced to logical propositions, originating instead in the intuitions of the mind.[82] Secondly, the notion of infinity as an expression of reality is itself disallowed in intuitionism, since the human mind cannot intuitively construct an infinite set.[83] Mathematicians such asL. E. J. Brouwer and especiallyHenri Poincaré adopted anintuitionist stance against Cantor's work. Finally,Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated theintension of a set of cardinal or real numbers with itsextension, thus conflating the concept of rules for generating a set with an actual set.[10]
Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.[6] In particular,neo-Thomist thinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity".[84] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:[85] "... the transfinite species are just as much at the disposal of the intentions of the Creator and His absolute boundless will as are the finite numbers.".[86] Prominent neo-scholastic German philosopher Konstantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God.[8]
Cantor also believed that his theory of transfinite numbers ran counter to bothmaterialism anddeterminism – and was shocked when he realized that he was the only faculty member at Halle who didnot hold to deterministic philosophical beliefs.[87]
It was important to Cantor that his philosophy provided an "organic explanation" of nature, and in his 1883Grundlagen, he said that such an explanation could only come about by drawing on the resources of the philosophy of Spinoza and Leibniz.[88] In making these claims, Cantor may have been influenced byF. A. Trendelenburg, whose lecture courses he attended at Berlin, and in turn Cantor produced a Latin commentary on Book 1 of Spinoza'sEthica. Trendelenburg was also the examiner of Cantor'sHabilitationsschrift.[89][90]
In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade other Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such asTilman Pesch andJoseph Hontheim,[91] as well as theologians such as CardinalJohann Baptist Franzelin, who once replied by equating the theory of transfinite numbers withpantheism.[7] Although later this Cardinal accepted the theory as valid, due to some clarifications from Cantor's.[8] Cantor even sent one letter directly toPope Leo XIII himself, and addressed several pamphlets to him.[85]
Cantor's philosophy on the nature of numbers led him to affirm a belief in the freedom of mathematics to posit and prove concepts apart from the realm of physical phenomena, as expressions within an internal reality. The only restrictions on thismetaphysical system are that all mathematical concepts must be devoid of internal contradiction, and that they follow from existing definitions, axioms, and theorems. This belief is summarized in his assertion that "the essence of mathematics is its freedom."[92] These ideas parallel those ofEdmund Husserl, whom Cantor had met in Halle.[93]
Meanwhile, Cantor himself was fiercely opposed toinfinitesimals, describing them as both an "abomination" and "thecholerabacillus of mathematics".[45]
Cantor's 1883 paper reveals that he was well aware of theopposition his ideas were encountering: "... I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers."[94]
Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free ofcontradiction and defined in terms of previously accepted concepts. He also cites Aristotle,René Descartes,George Berkeley,Gottfried Leibniz, andBernard Bolzano on infinity. Instead, he always strongly rejectedImmanuel Kant's philosophy, in the realms of both the philosophy of mathematics and metaphysics. He shared B. Russell's motto "Kant or Cantor", and defined Kant "yonder sophisticalPhilistine who knew so little mathematics."[95]
The title on the memorial plaque (in Russian): "In this building was born and lived from 1845 till 1854 the great mathematician and creator of set theory Georg Cantor",Vasilievsky Island, Saint-Petersburg.
Cantor's paternal grandparents were fromCopenhagen and fled to Russia from the disruption of theNapoleonic Wars. There is very little direct information on them.[96] Cantor's father, Georg Waldemar Cantor, was educated in theLutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. Very little is known for sure about Georg Waldemar's origin or education.[97] Cantor's mother, Maria Anna Böhm, was anAustro-Hungarian born in Saint Petersburg and baptizedRoman Catholic; she converted toProtestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, stating:
Mögen wir zehnmal von Juden abstammen und ich im Princip noch so sehr für Gleichberechtigung der Hebräer sein, im socialen Leben sind mir Christen lieber ... ("Even if we were descended from Jews ten times over, and even though I may be, in principle, completely in favour of equal rights for Hebrews, in social life I prefer Christians...")[97]
which could be read to imply that she was ofJewish ancestry.[98]
According to biographerEric Temple Bell, Cantor was of Jewish descent, although both parents were baptized.[99] In a 1971 article entitled "Towards a Biography of Georg Cantor", the British historian of mathematics Ivor Grattan-Guinness mentions (Annals of Science 27, pp. 345–391, 1971) that he was unable to find evidence of Jewish ancestry. (He also states that Cantor's wife, Vally Guttmann, was Jewish).
In a letter written toPaul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondence, Gauthier-Villars, Paris, 1934, p. 306), Cantor states that his paternal grandparents were members of the Sephardic Jewish community of Copenhagen. Specifically, Cantor states in describing his father: "Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde...." ("He was born in Copenhagen of Jewish (lit: 'Israelite') parents from the local Portuguese-Jewish community.")[100]In addition, Cantor's maternal great uncle,[101]Josef Böhm, a Hungarian violinist, has been described as Jewish,[102] which may imply that Cantor's mother was at least partly descended from the Hungarian Jewish community.[103]
In a letter toBertrand Russell, Cantor described his ancestry and self-perception as follows:
Neither my father nor my mother were of German blood, the first being a Dane, borne in Kopenhagen, my mother of Austrian Hungar descension. You must know, Sir, that I am not aregular just Germain, for I am born 3 March 1845 at Saint Peterborough, Capital of Russia, but I went with my father and mother and brothers and sister, eleven years old in the year 1856, into Germany.[104]
There were documented statements, during the 1930s, that called this Jewish ancestry into question:
More often [i.e., than the ancestry of the mother] the question has been discussed of whether Georg Cantor was of Jewish origin. About this it is reported in a notice of the Danish genealogical Institute in Copenhagen from the year 1937 concerning his father: "It is hereby testified that Georg Woldemar Cantor, born 1809 or 1814, is not present in the registers of the Jewish community, and that he completely without doubt was not a Jew ..."[97]
Until the 1970s, the chief academic publications on Cantor were two short monographs byArthur Moritz Schönflies (1927) – largely the correspondence with Mittag-Leffler – and Fraenkel (1930). Both were at second and third hand; neither had much on his personal life. The gap was largely filled byEric Temple Bell'sMen of Mathematics (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on thehistory of mathematics"; and as "one of the worst".[105] Bell presents Cantor's relationship with his father asOedipal, Cantor's differences with Kronecker as a quarrel between two Jews, and Cantor's madness as Romantic despair over his failure to win acceptance for his mathematics. Grattan-Guinness (1971) found that none of these claims were true, but they may be found in many books of the intervening period, owing to the absence of any other narrative. There are other legends, independent of Bell – including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents.[106] A critique of Bell's book is contained inJoseph Dauben's biography.[107] Writes Dauben:
Cantor devoted some of his most vituperative correspondence, as well as a portion of theBeiträge, to attacking what he described at one point as the 'infinitesimal Cholera bacillus of mathematics', which had spread from Germany through the work ofThomae,du Bois Reymond andStolz, to infect Italian mathematics ... Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete. Thus to accept the work of Thomae, du Bois-Reymond, Stolz andVeronese was to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible.[108]
^abDauben 1979, p. 280: "... the tradition made popular byArthur Moritz Schönflies blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression.
^Dauben 2004, p. 1. Text includes a 1964 quote from psychiatrist Karl Pollitt, one of Cantor's examining physicians at Halle Nervenklinik, referring to Cantor'smental illness as "cyclic manic-depression".
^Hilbert (1926, p. 170): "Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben können." (Literally: "Out of the Paradise that Cantor created for us, no one must be able to expel us.")
^O'Connor, JJ; Robertson, E F (October 1998)."Georg Ferdinand Ludwig Philipp Cantor".Maths History. University of St Andrews.Archived from the original on 9 February 2025. Retrieved9 February 2025.They married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind.
^O'Connor, JJ; Robertson, E F (October 1998)."Georg Ferdinand Ludwig Philipp Cantor".Maths History. University of St Andrews.Archived from the original on 9 February 2025. Retrieved9 February 2025.They married on 9 August 1874 and spent their honeymoon in Interlaken in Switzerland where Cantor spent much time in mathematical discussions with Dedekind.
^abJohnson, Phillip E. (1972). "The Genesis and Development of Set Theory".The Two-Year College Mathematics Journal.3 (1):55–62.doi:10.2307/3026799.JSTOR3026799.
^Suppes, Patrick (1972).Axiomatic Set Theory. Dover. p. 1.ISBN9780486616308.With a few rare exceptions the entities which are studied and analyzed in mathematics may be regarded as certain particular sets or classes of objects.... As a consequence, many fundamental questions about the nature of mathematics may be reduced to questions about set theory.
^Acountable set is a set which is either finite or denumerable; the denumerable sets are therefore the infinite countable sets. However, this terminology is not universally followed, and sometimes "denumerable" is used as a synonym for "countable".
^Cooke, Roger (1993). "Uniqueness of trigonometric series and descriptive set theory, 1870–1985".Archive for History of Exact Sciences.45 (4): 281.doi:10.1007/BF01886630.S2CID122744778.
^Cantor's construction starts with the set of transcendentalsT and removes a countablesubset {tn} (for example,tn = e / n). Call this setT0. ThenT =T0 ∪ {tn} =T0 ∪ {t2n-1} ∪ {t2n}. The set of realsR =T ∪ {an} =T0 ∪ {tn} ∪ {an} wherean is the sequence of real algebraic numbers. So bothT andR are the union of threepairwise disjoint sets:T0 and two countable sets. A one-to-one correspondence betweenT andR is given by the function:f(t) = t ift ∈ T0,f(t2n-1) = tn, andf(t2n) = an. Cantor actually applies his construction to the irrationals rather than the transcendentals, but he knew that it applies to any set formed by removing countably many numbers from the set of reals (Cantor 1879, p. 4).
^Dauben 1979, pp. 69, 32463n. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.
^Some mathematicians consider these results to have settled the issue, and, at most, allow that it is possible to examine the formal consequences of CH or of its negation, or of axioms that imply one of those. Others continue to look for "natural" or "plausible" axioms that, when added to ZFC, will permit either a proof or refutation of CH, or even for direct evidence for or against CH itself; among the most prominent of these isW. Hugh Woodin. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.
^Moore 1982, p. 51. Proof of equivalence: If a set is well-ordered, then its cardinality is an aleph since the alephs are the cardinals of well-ordered sets. If a set's cardinality is an aleph, then it can be well-ordered since there is a one-to-one correspondence between it and the well-ordered set defining the aleph.
^Cantor's proof, which is aproof by contradiction, starts by assuming there is a setS whose cardinality is not an aleph. A function from the ordinals toS is constructed by successively choosing different elements ofS for each ordinal. If this construction runs out of elements, then the function well-orders the setS. This implies that the cardinality ofS is an aleph, contradicting the assumption aboutS. Therefore, the function maps all the ordinals one-to-one intoS. The function'simage is an inconsistent submultiplicity contained inS, so the setS is an inconsistent multiplicity, which is a contradiction. Zermelo criticized Cantor's construction: "the intuition of time is applied here to a process that goes beyond all intuition, and a fictitious entity is posited of which it is assumed that it could makesuccessive arbitrary choices." (Hallett 1986, pp. 169–170.)
^Hallett 1986, pp. 288, 290–291. Cantor had pointed out that inconsistent multiplicities face the same restriction: they cannot be members of any multiplicity. (Hallett 1986, p. 286.)
^Dauben, Joseph Warren (1978)."Georg Cantor: The Personal Matrix of His Mathematics".Isis.69 (4): 548.doi:10.1086/352113.JSTOR231091.PMID387662.S2CID26155985. Retrieved5 March 2020.The religious dimension which Cantor attributed to his transfinite numbers should not be discounted as an aberration. Nor should it be forgotten or separated from his existence as a mathematician. The theological side of Cantor's set theory, though perhaps irrelevant for understanding its mathematical content, is nevertheless essential for the full understanding of his theory and why it developed in its early stages as it did.
^Davenport, Anne A. (1997). "The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century".Isis.88 (2):263–295.doi:10.1086/383692.JSTOR236574.S2CID154486558.
^For more information, see:Dauben 1979, p. 1 and notes;Grattan-Guinness 1971, pp. 350–352 and notes;Purkert and Ilgauds 1985; the letter is fromAczel 2000, pp. 93–94, from Louis' trip to Chicago in 1863. It is ambiguous in German, as in English, whether the recipient is included.
^Grattan-Guinness 1971 (quotation from p. 350, note),Dauben 1979, p. 1 and notes. (Bell's Jewish stereotypes appear to have been removed from some postwar editions.)
^Dauben, J.: The development of the Cantorian set theory, pp.~181–219. See pp.216–217. In Bos, H.; Bunn, R.; Dauben, J.;Grattan-Guinness, I.; Hawkins, T.; Pedersen, K. From the calculus to set theory, 1630–1910. An introductory history. Edited by I. Grattan-Guinness. Gerald Duckworth & Co. Ltd., London, 1980.
Dauben, Joseph (2004) [1993].Georg Cantor and the Battle for Transfinite Set Theory(PDF). Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, Calif.). pp. 1–22.Archived(PDF) from the original on 23 January 2018. Internet version published inJournal of the ACMS 2004. Note, though, that Cantor's Latin quotation described in this article asa familiar passage from the Bible is actually from the works of Seneca and has no implication of divine revelation.
Ewald, William B., ed. (1996).From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics. New York: Oxford University Press.ISBN978-0-19-853271-2.
Hallett, Michael (1986).Cantorian Set Theory and Limitation of Size. New York: Oxford University Press.ISBN978-0-19-853283-5.
Moore, Gregory H. (1982).Zermelo's Axiom of Choice: Its Origins, Development & Influence. Springer.ISBN978-1-4613-9480-8.
Moore, Gregory H. (19 April 2023)."The roots of Russell's paradox".Russell: The Journal of Bertrand Russell Archives.8 (1). Johns Hopkins University Press:46–56.ISSN1913-8032. Retrieved28 March 2025.
Suppes, Patrick (1972) [1960].Axiomatic Set Theory. New York: Dover.ISBN978-0-486-61630-8. Although the presentation is axiomatic rather than naive, Suppes proves and discusses many of Cantor's results, which demonstrates Cantor's continued importance for the edifice of foundational mathematics.
Aczel, Amir D. (2000).The Mystery of the Aleph: Mathematics, the Kabbala, and the Search for Infinity. New York: Four Walls Eight Windows Publishing..ISBN0-7607-7778-0. A popular treatment of infinity, in which Cantor is frequently mentioned.
Ferreirós, José (2007).Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought. Basel, Switzerland: Birkhäuser..ISBN3-7643-8349-6 Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory.
Hill, C. O.; Rosado Haddock, G. E. (2000).Husserl or Frege? Meaning, Objectivity, and Mathematics. Chicago: Open Court..ISBN0-8126-9538-0 Three chapters and 18 index entries on Cantor.
Meschkowski, Herbert (1983).Georg Cantor, Leben, Werk und Wirkung (Georg Cantor, Life, Work and Influence, in German). Vieweg, Braunschweig.
Newstead, Anne (2009). "Cantor on Infinity in Nature, Number, and the Divine Mind"[2],American Catholic Philosophical Quarterly,83 (4): 532–553,https://doi.org/10.5840/acpq200983444. With acknowledgement of Dauben's pioneering historical work, this article further discusses Cantor's relation to the philosophy of Spinoza and Leibniz in depth, and his engagement in thePantheismusstreit. Brief mention is made of Cantor's learning from F.A.Trendelenburg.
Rucker, Rudy (2005) [1982].Infinity and the Mind. Princeton University Press..ISBN0-553-25531-2 Deals with similar topics to Aczel, but in more depth.
Rodych, Victor (2007)."Wittgenstein's Philosophy of Mathematics". In Edward N. Zalta (ed.).The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University..
Leonida Lazzari,L'infinito di Cantor. Editrice Pitagora, Bologna, 2008.
"Cantor infinities", analysis of Cantor's 1874 article,BibNum(for English version, click 'à télécharger'). There is an error in this analysis. It states Cantor's Theorem 1 correctly: Algebraic numbers can be counted. However, it states his Theorem 2 incorrectly: Real numbers cannot be counted. It then says: "Cantor notes that, taken together, Theorems 1 and 2 allow for the redemonstration of the existence of non-algebraic real numbers …" This existence demonstration isnon-constructive. Theorem 2 stated correctly is: Given a sequence of real numbers, one can determine a real number that is not in the sequence. Taken together, Theorem 1 and this Theorem 2 produce a non-algebraic number. Cantor also used Theorem 2 to prove that the real numbers cannot be counted. SeeCantor's first set theory article orGeorg Cantor and Transcendental NumbersArchived 21 January 2022 at theWayback Machine.
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