Georg Cantor

Georg Cantor
Cantor,c. 1910
Born	Georg Ferdinand Ludwig Philipp Cantor (1845-03-03)3 March 1845 Saint Petersburg,Russian Empire
Died	6 January 1918(1918-01-06)(aged 72) Halle,Province of Saxony, German Empire
Nationality	German-Russian
Alma mater	Swiss Federal Polytechnic University of Berlin University of Göttingen
Known for	Set theory
Spouse	Vally Guttmann ​ (m.1874)​
Awards	Sylvester Medal(1904)
Scientific career
Fields	Mathematics
Institutions	University of Halle
Thesis	De aequationibus secundi gradus indeterminatis(1867)
Doctoral advisor	Ernst Kummer Karl Weierstrass
Signature

Georg Ferdinand Ludwig Philipp Cantor(/ˈkæntɔːr/KAN-tor,German:[ˈɡeːɔʁkˈfɛʁdinantˈluːtvɪçˈfiːlɪpˈkantoːɐ̯];3 March [O.S.19 February] 1845 – 6 January 1918) was a mathematician who played a pivotal role in the creation ofset theory,which has become afundamental theoryin mathematics. Cantor established the importance ofone-to-one correspondencebetween the members of two sets, definedinfiniteandwell-ordered sets,and proved that thereal numbersare more numerous than thenatural numbers.Cantor's method of proof of this theorem implies the existence of aninfinityof infinities. He defined thecardinalandordinalnumbers and their arithmetic. Cantor's work is of great philosophical interest, a fact he was well aware of.

Originally, Cantor's theory oftransfinite numberswas regarded as counter-intuitive – even shocking. This caused it to encounter resistance from mathematical contemporaries such asLeopold KroneckerandHenri Poincaréand later fromHermann WeylandL. E. J. Brouwer,whileLudwig Wittgensteinraisedphilosophical objections;seeControversy over Cantor's theory.Cantor, a devoutLutheran Christian,believed the theory had been communicated to him by God.SomeChristian theologians(particularlyneo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God– on one occasion equating the theory of transfinite numbers withpantheism– a proposition that Cantor vigorously rejected. Not all theologians were against Cantor's theory; prominent neo-scholastic philosopher Constantin Gutberlet was in favor of it and CardinalJohann Baptist Franzelinaccepted it as a valid theory (after Cantor made some important clarifications).

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".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".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,though some have explained these episodes as probable manifestations of abipolar disorder.

The harsh criticism has been matched by later accolades. In 1904, theRoyal Societyawarded Cantor itsSylvester Medal,the highest honor it can confer for work in mathematics.David Hilbertdefended it from its critics by declaring, "No one shall expel us from theparadise that Cantor has created."

Biography

Youth and studies

Cantor, around 1870

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 grandfather Franz Böhm (1788–1846) (the violinistJoseph Böhm's brother) was a well-known musician and soloist in a Russian imperial orchestra.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,trigonometryin particular, were noted. In August 1862, he then graduated from the "Höhere Gewerbeschule Darmstadt", now theTechnische Universität Darmstadt.In 1862 Cantor entered theSwiss Federal Polytechnicin Zurich. After receiving a substantial inheritance upon his father's death in June 1863,Cantor transferred to theUniversity of Berlin,attending lectures byLeopold Kronecker,Karl WeierstrassandErnst 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.

Teacher and researcher

Cantor submitted hisdissertationonnumber theoryat 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 requisitehabilitationfor his thesis, also on number theory, which he presented in 1869 upon his appointment atHalle University.

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 theHarz mountains,Cantor spent much time in mathematical discussions withRichard Dedekind,whom he had met at Interlaken in Switzerland two years earlier while on holiday.

Cantor was promoted to extraordinary professor in 1872 and made full professor in 1879.To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired achairat 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.Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.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.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,so Cantor came to believe that Kronecker's stance would make it impossible for him ever to leave Halle.

In 1881, Cantor's Halle colleagueEduard Heinedied. Halle accepted Cantor's suggestion that Heine's vacant chair be offered to Dedekind,Heinrich M. WeberandFranz 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.

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.Cantor also began another important correspondence, withGösta Mittag-Lefflerin 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.He asked Cantor to withdraw the paper fromActawhile 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."

Cantor suffered his first known bout of depression in May 1884.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.

This crisis led him to apply to lecture on philosophy rather than on mathematics. He also began an intense study ofElizabethan literature,thinking there might be evidence thatFrancis Baconwrote the plays attributed toWilliam Shakespeare(seeShakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.

Cantor recovered soon thereafter, and subsequently made further important contributions, including hisdiagonal argumentandtheorem.However, he never again attained the high level of his remarkable papers of 1874–84, even after Kronecker's death on December 29, 1891.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,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.

Later years and death

After Cantor's 1884 hospitalization there is no record that he was in anysanatoriumagain until 1899.Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly on December 16 (Cantor was delivering a lecture on his views onBaconian theoryandWilliam Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics.Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented byJulius Königat the ThirdInternational Congress of Mathematicians.The paper attempted to prove that the basic tenets oftransfinite set theorywere false. Since the paper had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated.AlthoughErnst Zermelodemonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God.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.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-Vereinigungin 1903, and attending the International Congress of Mathematicians atHeidelbergin 1904.

In 1911, Cantor was one of the distinguished foreign scholars invited to the 500th anniversary of the founding of theUniversity of St. Andrewsin Scotland. Cantor attended, hoping to meetBertrand Russell,whose newly publishedPrincipia Mathematicarepeatedly 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 frommalnourishmentduringWorld War I.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 January 6, 1918, in the sanatorium where he had spent the last year of his life.

Mathematical work

Cantor's work between 1874 and 1884 is the origin ofset theory.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 theoryhas come to play the role of afoundational theoryin 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.

In one of his earliest papers,Cantor proved that the set ofreal numbersis "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 definefiniteandinfinite sets,subdividing the latter intodenumerable(or countably infinite) sets andnondenumerable sets(uncountably infinite sets).

Cantor developed important concepts intopologyand their relation tocardinality.For example, he showed that theCantor set,discovered byHenry John Stephen Smithin 1875,isnowhere dense,but has the same cardinality as the set of all real numbers, whereas therationalsare everywhere dense, but countable. He also showed that all countable denselinear orderswithout end points are order-isomorphic to therational numbers.

Cantor introduced fundamental constructions in set theory, such as thepower setof a setA,which is the set of all possiblesubsetsofA.He later proved that the size of the power set ofAis strictly larger than the size ofA,even whenAis an infinite set; this result soon became known asCantor's theorem.Cantor developed an entire theory andarithmetic of infinite sets,calledcardinalsandordinals,which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter$\aleph$(ℵ,aleph) with a natural number subscript; for the ordinals he employed the Greek letter$\omega$(ω,omega). This notation is still in use today.

TheContinuum hypothesis,introduced by Cantor, was presented byDavid Hilbertas the first of histwenty-three open problemsin his address at the 1900International Congress of Mathematiciansin Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium.The US philosopherCharles Sanders Peircepraised Cantor's set theory and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zürich in 1897,Adolf HurwitzandJacques Hadamardalso 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 Jourdainon the history ofset theoryand on Cantor's religious ideas. This was later published, as were several of his expository works.

Number theory, trigonometric series and ordinals

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 problemthat had eludedPeter Gustav Lejeune Dirichlet,Rudolf Lipschitz,Bernhard Riemann,and Heine himself: the uniqueness of the representation of afunctionbytrigonometric series.Cantor solved this problem in 1869. It was while working on this problem that he discovered transfinite ordinals, which occurred as indicesnin thenthderived setS_nof a setSof zeros of a trigonometric series. Given a trigonometric series f(x) withSas its set of zeros, Cantor had discovered a procedure that produced another trigonometric series that hadS₁as its set of zeros, whereS₁is the set oflimit pointsofS.IfS_k+1is the set of limit points ofS_k,then he could construct a trigonometric series whose zeros areS_k+1.Because the setsS_kwere 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 callS_ω,and then he noticed thatS_ωwould also have to have a set of limit pointsS_ω+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,...

Between 1870 and 1872, Cantor published more papers on trigonometric series, and also a paper definingirrational numbersasconvergent sequencesofrational 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 ofinfinitesimalsof his contemporariesOtto StolzandPaul du Bois-Reymond,describing them as both "an abomination" and "acholerabacillusof mathematics ".Cantor also published an erroneous "proof" of the inconsistency ofinfinitesimals.

Set theory

An illustration ofCantor's diagonal argumentfor the existence ofuncountable sets.The sequence at the bottom cannot occur anywhere in the infinite list of sequences above.

The beginning of set theory as a branch of mathematics is often marked by the publication ofCantor's 1874 paper,"Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ( "On a Property of the Collection of All Real Algebraic Numbers" ).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).Cantor proved that the collection of real numbers and the collection of positiveintegersare not equinumerous. In other words, the real numbers are notcountable.His proof differs from thediagonal argumentthat he gave in 1891.Cantor's article also contains a new method of constructingtranscendental numbers.Transcendental numbers were first constructed byJoseph Liouvillein 1844.

Cantor established these results using two constructions. His first construction shows how to write the realalgebraic numbersas 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 intervalswhoseintersectioncontains 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.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.

Between 1879 and 1884, Cantor published a series of six articles inMathematische Annalenthat 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 afinitenumber 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 infinitywould open the door to paradoxes which would challenge the validity of mathematics as a whole.Cantor also introduced theCantor setduring this period.

The fifth paper in this series, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre "( "Foundations of a General Theory of Aggregates "), published in 1883,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 numberswere a systematic extension of the natural numbers. It begins by definingwell-orderedsets.Ordinal numbersare then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of thecardinaland 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:thecardinalityof the power set of a setAis strictly larger than the cardinality ofA.This established the richness of the hierarchy of infinite sets, and of thecardinalandordinal arithmeticthat Cantor had defined. His argument is fundamental in the solution of theHalting problemand the proof ofGödel's first incompleteness theorem.Cantor wrote on theGoldbach conjecturein 1894.

Passage of Georg Cantor's article with his set definition

In 1895 and 1897, Cantor published a two-part paper inMathematische AnnalenunderFelix Klein's editorship; these were his last significant papers on set theory.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 setsand ordinal numbers. Cantor attempts to prove that ifAandBare sets withAequivalentto a subset ofBandBequivalent to a subset ofA,thenAandBare equivalent.Ernst Schröderhad stated this theorem a bit earlier, but his proof, as well as Cantor's, was flawed.Felix Bernsteinsupplied a correct proof in his 1898 PhD thesis; hence the nameCantor–Bernstein–Schröder theorem.

One-to-one correspondence

Main article:Bijection

A bijective function

Cantor's 1874Crellepaper 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 squareand the points of a unitline segment.In an 1877 letter to Richard Dedekind, Cantor proved a farstrongerresult: 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! ")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 spaceRⁿhas the same power as thereal numbersR,as does a countably infiniteproductof 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 hismappingbetween theunit intervaland the unit square was not acontinuousone.

This paper displeased Kronecker and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so andKarl Weierstrasssupported its publication.Nevertheless, Cantor never again submitted anything to Crelle.

Continuum hypothesis

Main article:Continuum hypothesis

Cantor was the first to formulate what later came to be known as thecontinuum hypothesisor 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 isexactlyaleph-one, rather than justat leastaleph-one). Cantor believed the continuum hypothesis to be true and tried for many years toproveit, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result byKurt Gödeland a 1963 one byPaul Cohentogether imply that the continuum hypothesis can be neither proved nor disproved using standardZermelo–Fraenkel set theoryplus theaxiom of choice(the combination referred to as "ZFC").

Absolute infinite, well-ordering theorem, and paradoxes

In 1883, Cantor divided the infinite into the transfinite and theabsolute.

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.In 1883, Cantor also introduced thewell-ordering principle"every set can be well-ordered" and stated that it is a "law of thought".

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.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.In 1932, Zermelo criticized the construction in Cantor's proof.

Cantor avoidedparadoxesby 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 Russelltreated 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.Russell named paradoxes afterCesare Burali-Fortiand Cantor even though neither of them believed that they had found paradoxes.

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.Zermelo had proved this theorem in 1904 using theaxiom of choice,but his proof was criticized for a variety of reasons.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.

In 1923,John von Neumanndeveloped 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 aclassis 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.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.In 1930, Zermelo definedmodels of set theory that satisfy von Neumann's axiom.

Philosophy, religion, literature and Cantor's mathematics

The concept of the existence of anactual infinitywas an important shared concern within the realms of mathematics, philosophy and religion. Preserving theorthodoxyof the relationship between God and mathematics, although not in the same form as held by his critics, was long a concern of Cantor's.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.To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications – he identified theabsolute infinitewith God,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.He was a devout Lutheran whose explicit Christian beliefs shaped his philosophy of science.Joseph Daubenhas traced the effect Cantor's Christian convictions had on the development of transfinite set theory.

Debate among mathematicians grew out of opposing views in thephilosophy of mathematicsregarding the nature of actual infinity. Some held to the view that infinity was an abstraction which was not mathematically legitimate, and denied its existence.Mathematicians from three major schools of thought (constructivismand its two offshoots,intuitionismandfinitism) 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 proofssuch as Cantor's diagonal argument are sufficient proof that something exists, holding instead thatconstructive proofsare 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.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.Mathematicians such asL. E. J. Brouwerand especiallyHenri Poincaréadopted anintuitioniststance against Cantor's work. Finally,Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated theintensionof a set of cardinal or real numbers with itsextension,thus conflating the concept of rules for generating a set with an actual set.

Some Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.In particular,neo-Thomistthinkers saw the existence of an actual infinity that consisted of something other than God as jeopardizing "God's exclusive claim to supreme infinity".Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:"... 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.".Prominent neo-scholastic German philosopher Constantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God.

Cantor also believed that his theory of transfinite numbers ran counter to bothmaterialismanddeterminism– and was shocked when he realized that he was the only faculty member at Halle who didnothold to deterministic philosophical beliefs.

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.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.

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 PeschandJoseph Hontheim,as well as theologians such as CardinalJohann Baptist Franzelin,who once replied by equating the theory of transfinite numbers withpantheism.Although later this Cardinal accepted the theory as valid, due to some clarifications from Cantor's.Cantor even sent one letter directly toPope Leo XIIIhimself, and addressed several pamphlets to him.

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 thismetaphysicalsystem 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."These ideas parallel those ofEdmund Husserl,whom Cantor had met in Halle.

Meanwhile, Cantor himself was fiercely opposed toinfinitesimals,describing them as both an "abomination" and "thecholerabacillusof mathematics ".

Cantor's 1883 paper reveals that he was well aware of theoppositionhis 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."

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free ofcontradictionand defined in terms of previously accepted concepts. He also cites Aristotle,René Descartes,George Berkeley,Gottfried Leibniz,andBernard Bolzanoon 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 sophisticalPhilistinewho knew so little mathematics. "

Cantor's ancestry

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 fromCopenhagenand fled to Russia from the disruption of theNapoleonic Wars.There is very little direct information on them.Cantor's father, Georg Waldemar Cantor, was educated in theLutheranmission 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.Cantor's mother, Maria Anna Böhm, was anAustro-Hungarianborn in Saint Petersburg and baptizedRoman Catholic;she converted toProtestantismupon 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..." ) which could be read to imply that she was of Jewish ancestry.

According to biographersEric Temple Bell,Cantor was of Jewish descent, although both parents were baptized.In a 1971 article entitled "Towards a Biography of Georg Cantor", the British historian of mathematics Ivor Grattan-Guinness mentions (Annals of Science27, 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 Tanneryin 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." ) In addition, Cantor's maternal great uncle,a Hungarian violinistJosef Böhm,has been described as Jewish,which may imply that Cantor's mother was at least partly descended from the Hungarian Jewish community.

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.

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..."

Biographies

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 ".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.A critique of Bell's book is contained inJoseph Dauben's biography.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 'infinitesimalCholera bacillus of mathematics', which had spread from Germany through the work ofThomae,du Bois ReymondandStolz,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 andVeronesewas to deny the perfection of Cantor's own creation. Understandably, Cantor launched a thorough campaign to discredit Veronese's work in every way possible.

See also

• Absolute infinite
• Aleph number
• Cardinality of the continuum
• Cantor medal– award by theDeutsche Mathematiker-Vereinigungin honor of Georg Cantor
• Cardinal number
• Continuum hypothesis
• Countable set
• Derived set (mathematics)
• Epsilon numbers (mathematics)
• Factorial number system
• Pairing function
• Transfinite number
• List of things named after Georg Cantor

References

• Dauben, Joseph W.(1977). "Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite".Journal of the History of Ideas.38(1): 85–108.doi:10.2307/2708842.JSTOR2708842..
• Dauben, Joseph W. (1979).[Unavailable on archive.org] Georg Cantor: his mathematics and philosophy of the infinite.Boston: Harvard University Press.ISBN978-0-691-02447-9..
• 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 ACMS2004. Note, though, that Cantor's Latin quotation described in this article asa familiar passage from the Bibleis 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..
• Grattan-Guinness, Ivor(1971). "Towards a Biography of Georg Cantor".Annals of Science.27(4): 345–391.doi:10.1080/00033797100203837..
• Grattan-Guinness, Ivor(2000).The Search for Mathematical Roots: 1870–1940.Princeton University Press.ISBN978-0-691-05858-0..
• 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. (1988)."The Roots of Russell's Paradox".Russell: The Journal of Bertrand Russell Studies.8:46–56.doi:10.15173/russell.v8i1.1732..
• Moore, Gregory H.; Garciadiego, Alejandro (1981)."Burali-Forti's Paradox: A Reappraisal of Its Origins".Historia Mathematica.8(3): 319–350.doi:10.1016/0315-0860(81)90070-7..
• Purkert, Walter (1989). "Cantor's Views on theFoundations of Mathematics".In Rowe, David E.; McCleary, John (eds.).The History of Modern Mathematics, Volume 1.Academic Press. pp.49–65.ISBN978-0-12-599662-4..
• Purkert, Walter; Ilgauds, Hans Joachim (1985).Georg Cantor: 1845–1918.Birkhäuser.ISBN978-0-8176-1770-7..
• 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.
• Zermelo, Ernst (1908)."Untersuchungen über die Grundlagen der Mengenlehre I".Mathematische Annalen.65(2): 261–281.doi:10.1007/bf01449999.S2CID120085563..
• Zermelo, Ernst (1930)."Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre"(PDF).Fundamenta Mathematicae.16:29–47.doi:10.4064/fm-16-1-29-47.Archived(PDF)from the original on 28 June 2004..
• van Heijenoort, Jean (1967).From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931.Harvard University Press.ISBN978-0-674-32449-7..

Bibliography

Older sources on Cantor's life should be treated with caution. See section§ Biographiesabove.

Primary literature in English

• Cantor, Georg (1955) [1915].Philip Jourdain(ed.).Contributions to the Founding of the Theory of Transfinite Numbers.New York: Dover Publications.ISBN978-0-486-60045-1..

Primary literature in German

• Cantor, Georg (1874)."Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen"(PDF).Journal für die Reine und Angewandte Mathematik.1874(77): 258–262.doi:10.1515/crll.1874.77.258.S2CID199545885.Archived(PDF)from the original on 7 October 2017.
• Cantor, Georg (1878)."Ein Beitrag zur Mannigfaltigkeitslehre".Journal für die Reine und Angewandte Mathematik.1878(84): 242–258.doi:10.1515/crelle-1878-18788413..
• Georg Cantor (1879)."Ueber unendliche, lineare Punktmannichfaltigkeiten (1)".Mathematische Annalen.15(1): 1–7.doi:10.1007/bf01444101.S2CID179177510.
• Georg Cantor (1880)."Ueber unendliche, lineare Punktmannichfaltigkeiten (2)".Mathematische Annalen.17(3): 355–358.doi:10.1007/bf01446232.S2CID179177438.
• Georg Cantor (1882)."Ueber unendliche, lineare Punktmannichfaltigkeiten (3)".Mathematische Annalen.20(1): 113–121.doi:10.1007/bf01443330.S2CID177809016.
• Georg Cantor (1883)."Ueber unendliche, lineare Punktmannichfaltigkeiten (4)".Mathematische Annalen.21(1): 51–58.doi:10.1007/bf01442612.S2CID179177480.
• Georg Cantor (1883)."Ueber unendliche, lineare Punktmannichfaltigkeiten (5)".Mathematische Annalen.21(4): 545–591.doi:10.1007/bf01446819.S2CID121930608.Published separately as:Grundlagen einer allgemeinen Mannigfaltigkeitslehre.
• Georg Cantor (1884)."Ueber unendliche, lineare Punktmannichfaltigkeiten (6)".Mathematische Annalen.23(4): 453–488.doi:10.1007/BF01446598.S2CID179178052.
• Georg Cantor (1891)."Ueber eine elementare Frage der Mannigfaltigkeitslehre"(PDF).Jahresbericht der Deutschen Mathematiker-Vereinigung.1:75–78.Archived(PDF)from the original on 1 January 2018.
• Cantor, Georg (1895)."Beiträge zur Begründung der transfiniten Mengenlehre (1)".Mathematische Annalen.46(4): 481–512.doi:10.1007/bf02124929.S2CID177801164.Archived fromthe originalon 23 April 2014.
• Cantor, Georg (1897)."Beiträge zur Begründung der transfiniten Mengenlehre (2)".Mathematische Annalen.49(2): 207–246.doi:10.1007/bf01444205.S2CID121665994.
• Cantor, Georg (1932).Ernst Zermelo(ed.)."Gesammelte Abhandlungen mathematischen und philosophischen inhalts".Berlin: Springer. Archived fromthe originalon February 3, 2014..Almost everything that Cantor wrote. Includes excerpts of his correspondence withDedekind(p. 443–451) andFraenkel'sCantor biography (p. 452–483) in the appendix.

Secondary literature

• 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.
• Dauben, Joseph W. (June 1983). "Georg Cantor and the Origins of Transfinite Set Theory".Scientific American.248(6): 122–131.Bibcode:1983SciAm.248f.122D.doi:10.1038/scientificamerican0683-122.
• Ferreirós, José (2007).Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought.Basel, Switzerland: Birkhäuser..ISBN3-7643-8349-6Contains a detailed treatment of both Cantor's and Dedekind's contributions to set theory.
• Halmos, Paul(1998) [1960].Naive Set Theory.New York & Berlin: Springer..ISBN3-540-90092-6
• Hilbert, David(1926)."Über das Unendliche".Mathematische Annalen.95:161–190.doi:10.1007/BF01206605.S2CID121888793.
• Hill, C. O.; Rosado Haddock, G. E. (2000).Husserl or Frege? Meaning, Objectivity, and Mathematics.Chicago: Open Court..ISBN0-8126-9538-0Three 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"[1],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.
• Penrose, Roger(2004).The Road to Reality.Alfred A. Knopf..ISBN0-679-77631-1Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporarytheoretical physicist.
• Rucker, Rudy(2005) [1982].Infinity and the Mind.Princeton University Press..ISBN0-553-25531-2Deals 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.

External links

• Quotations related toGeorg Cantorat Wikiquote
• Media related toGeorg Cantorat Wikimedia Commons
• Works by or about Georg CantoratInternet Archive
• O'Connor, John J.;Robertson, Edmund F.,"Georg Cantor",MacTutor History of Mathematics Archive,University of St Andrews
• O'Connor, John J.;Robertson, Edmund F.,"A history of set theory",MacTutor History of Mathematics Archive,University of St AndrewsMainly devoted to Cantor's accomplishment.
• Georg Cantor,britannica.com
• Stanford Encyclopedia of Philosophy:Set theorybyThomas Jech.The Early Development of Set Theoryby José Ferreirós.
• "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 articleorGeorg Cantor and Transcendental Numbers.