# Georg Cantor: Wikis

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# Encyclopedia

Georg Cantor

Born Georg Ferdinand Ludwig Philipp Cantor
March 3, 1845
Saint Petersburg, Russia
Died January 6, 1918 (aged 72)
Halle, Germany
Residence Russia (1845–1856),
Germany (1856–1918)
Fields Mathematics
Institutions University of Halle
Alma mater ETH Zurich, University of Berlin
Karl Weierstrass
Doctoral students Alfred Barneck
Known for Set theory

Georg Ferdinand Ludwig Philipp Cantor (March 3 [O.S. February 19] 1845[1] – January 6, 1918) was a mathematician, best known as the creator of set theory, which has become a fundamental theory in mathematics. Cantor established the importance of one-to-one correspondence between sets, defined infinite and well-ordered sets, and proved that the real numbers are "more numerous" than the natural numbers. In fact, Cantor's theorem implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.[2]

Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even shocking—that it encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God,[4] on one occasion equating the theory of transfinite numbers with pantheism.[5] The objections to his work were occasionally fierce: Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,[6] and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific charlatan", a "renegade" and a "corrupter of youth."[7] 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".[8] Cantor's recurring bouts of depression from 1884 to the end of his life were once blamed on the hostile attitude of many of his contemporaries,[9] but these episodes can now be seen as probable manifestations of a bipolar disorder.[10]

The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.[11] Cantor believed his theory of transfinite numbers had been communicated to him by God.[12] David Hilbert defended it from its critics by famously declaring: "No one shall expel us from the Paradise that Cantor has created."[13]

## Life

### Youth and studies

Cantor was born in 1845 in the western merchant colony in Saint Petersburg, Russia, and brought up in the city until he was eleven. Georg, the eldest of six children, was an outstanding violinist, having inherited his parents' considerable musical and artistic talents. Cantor's father had been a member of the Saint Petersburg stock exchange; when he became ill, the family moved to Germany in 1856, first to Wiesbaden then to Frankfurt, seeking winters milder than those of Saint Petersburg. In 1860, Cantor graduated with distinction from the Realschule in Darmstadt; his exceptional skills in mathematics, trigonometry in particular, were noted. In 1862, Cantor entered the Federal Polytechnic Institute in Zürich, today the ETH Zurich. After receiving a substantial inheritance upon his father's death in 1863, Cantor shifted his studies to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer. He spent the summer of 1866 at the University of Göttingen, then and later a very important center for mathematical research. In 1867, Berlin granted him the PhD for a thesis on number theory, De aequationibus secundi gradus indeterminatis.

### Teacher and researcher

After teaching briefly in a Berlin girls' school, Cantor took up a position at the University of Halle, where he spent his entire career. He was awarded the requisite habilitation for his thesis on number theory.

In 1874, Cantor married Vally Guttmann. They had six children, the last (Rudolph) born in 1886. Cantor was able to support a family despite modest academic pay, thanks to his inheritance from his father. During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he befriended two years earlier while on Swiss 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 a chair at a more prestigious university, in particular at Berlin, then the leading German university. However, his work encountered too much opposition for that to be possible.[14] Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,[15] perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.[16] Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, fundamentally disagreed with the thrust of Cantor's work. Kronecker, now seen as one of the founders of the constructive 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. Cantor came to believe that Kronecker's stance would make it impossible for Cantor to ever leave Halle.

In 1881, Cantor's Halle colleague Eduard Heine died, creating a vacant chair. Halle accepted Cantor's suggestion that it be offered to Dedekind, Heinrich M. Weber and Franz 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 refusal to accept the chair at Halle.[17] Cantor also began another important correspondence, with Gösta Mittag-Leffler in Sweden, and soon began to publish in Mittag-Leffler's journal Acta Mathematica. But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta.[18] He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "... about one hundred years too soon." Cantor complied, but wrote 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 about Acta Mathematica."[19] ”

Cantor then sharply curtailed his relationship and correspondence with Mittag-Leffler, displaying a tendency to interpret well-intentioned criticism as a deeply personal affront.

Cantor suffered his first known bout of depression in 1884.[20] Criticism of his work weighed on his mind: every one of the fifty-two letters he wrote to Mittag-Leffler in 1884 attacked 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."[21] ”

This emotional crisis led him to apply to lecture on philosophy rather than mathematics. He also began an intense study of Elizabethan literature in an attempt to prove that Francis Bacon wrote the plays attributed to Shakespeare (see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.[22]

Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem. However, he never again attained the high level of his remarkable papers of 1874–1884. He eventually sought a reconciliation with Kronecker, which Kronecker graciously accepted. Nevertheless, the philosophical disagreements and difficulties dividing them persisted. It was once thought that Cantor's recurring bouts of depression were triggered by the opposition his work met at the hands of Kronecker.[9] While Cantor's mathematical worries and his difficulties dealing with certain people were greatly magnified by his depression, it is doubtful that they were its cause. Rather, his posthumous diagnosis of bipolarity has been accepted as the root cause of his erratic mood.[10]

In 1890, Cantor was instrumental in founding the Deutsche Mathematiker-Vereinigung and 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 he felt towards Kronecker, Cantor invited him to address the meeting, but Kronecker was unable to do so because his spouse was dying from a skiing accident at the time.

### Late years

After Cantor's 1884 hospitalization, there is no record that he was in any sanatorium again until 1899.[20] Soon after that second hospitalization, Cantor's youngest son Rudolph died suddenly (while Cantor was delivering a lecture on his views on Baconian theory and William Shakespeare), and this tragedy drained Cantor of much of his passion for mathematics.[23] Cantor was again hospitalized in 1903. One year later, he was outraged and agitated by a paper presented by Julius König at the Third International Congress of Mathematicians. The paper attempted to prove that the basic tenets of transfinite set theory were false. Since it had been read in front of his daughters and colleagues, Cantor perceived himself as having been publicly humiliated.[24] Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, even momentarily questioning God.[11] Cantor suffered from chronic depression for the rest of his life, for which he was excused from teaching on several occasions and repeatedly confined in various sanatoria. The events of 1904 preceded a series of hospitalizations at intervals of two or three years.[25] He did not abandon mathematics completely, however, lecturing on the paradoxes of set theory (Burali-Forti paradox, Cantor's paradox, and Russell's paradox) to a meeting of the Deutsche Mathematiker–Vereinigung in 1903, and attending the International Congress of Mathematicians at Heidelberg in 1904.

In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland. Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but this 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 suffered from poverty, even malnourishment, during World War I.[26] The public celebration of his 70th birthday was canceled because of the war. He died on January 6, 1918 in the sanatorium where he had spent the final year of his life.

## Mathematical work

Cantor's work between 1874 and 1884 is the origin of set theory.[27] Prior to this work, the concept of a set was a rather elementary one that had been used implicitly since the beginnings of mathematics, dating back to the ideas of Aristotle.[28] 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 a foundational 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 as algebra, analysis and topology) 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 of real numbers is "more numerous" than the set of natural numbers; this showed, for the first time, that there exist infinite sets of different sizes. He was also the first to appreciate the importance of one-to-one correspondences (hereinafter denoted "1-to-1") in set theory. He used this concept to define finite and infinite sets, subdividing the latter into denumerable (or countably infinite) sets and uncountable sets (nondenumerable infinite sets).[29]

Cantor introduced fundamental constructions in set theory, such as the power set of a set A, which is the set of all possible subsets of A. He later proved that the size of the power set of A is strictly larger than the size of A, even when A is an infinite set; this result soon became known as Cantor's theorem. Cantor developed an entire theory and arithmetic of infinite sets, called cardinals and ordinals, which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter $\aleph$ (aleph) with a natural number subscript; for the ordinals he employed the Greek letter ω (omega). This notation is still in use today.

The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his famous address at the 1900 International Congress of Mathematicians in Paris. Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium.[30] The US philosopher Charles Sanders Peirce praised Cantor's set theory, and, following public lectures delivered by Cantor at the first International Congress of Mathematicians, held in Zurich in 1897, Hurwitz and 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 translator Philip Jourdain on the history of set theory and on Cantor's religious ideas. This was later published, as were several of his expository works.

### Number theory and function theory

Cantor's first ten papers were on number theory, his thesis topic. At the suggestion of Eduard Heine, the Professor at Halle, Cantor turned to analysis. Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series. Cantor solved this difficult problem in 1869. Between 1870 and 1872, Cantor published more papers on trigonometric series, including one defining irrational numbers as convergent sequences of rational 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 by Dedekind cuts.

### Set theory

An illustration of Cantor's diagonal argument for the existence of uncountable sets.[31] 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 of Cantor's 1874 paper, "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Characteristic Property of All Real Algebraic Numbers").[27] The paper, published in Crelle's Journal thanks to Dedekind's support (and despite Kronecker's opposition), was the first to formulate a mathematically rigorous proof that there was more than one kind of infinity. This demonstration is a centerpiece of his legacy as a mathematician, helping lay the groundwork for both calculus and the analysis of the continuum of real numbers.[32] Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements).[33] He then proved that the real numbers were not countable, albeit employing a proof more complex than the remarkably elegant and justly celebrated diagonal argument he set out in 1891.[34] Prior to this, he had already proven that the set of rational numbers is countable.

Joseph Liouville had established the existence of transcendental numbers in 1844,[35] and Cantor's paper established that the set of transcendental numbers is uncountable. The logic is as follows: Cantor had shown that the union of two countable sets must be countable. The set of all real numbers is equal to the union of the set of algebraic numbers with the set of transcendental numbers (that is, every real number must be either algebraic or transcendental). The 1874 paper showed that the algebraic numbers (that is, the roots of polynomial equations with integer coefficients), were countable. In contrast, Cantor had also just shown that the real numbers were not countable. If transcendental numbers were countable, then the result of their union with algebraic numbers would also be countable. Since their union (which equals the set of all real numbers) is uncountable, it logically follows that the transcendentals must be uncountable. The transcendentals have the same "power" (see below) as the reals, and "almost all" real numbers must be transcendental. Cantor remarked that he had effectively reproved a theorem, due to Liouville, to the effect that there are infinitely many transcendental numbers in each interval.

Between 1879 and 1884, Cantor published a series of six articles in Mathematische Annalen that together formed an introduction to his set theory. At the same time, there was growing opposition to Cantor's ideas, led by Kronecker, who admitted mathematical concepts only if they could be constructed in a finite 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 of actual infinity would open the door to paradoxes which would challenge the validity of mathematics as a whole.[36] Cantor also discovered the Cantor set during 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 separate monograph. It contained Cantor's reply to his critics and showed how the transfinite numbers were a systematic extension of the natural numbers. It begins by defining well-ordered sets. Ordinal numbers are then introduced as the order types of well-ordered sets. Cantor then defines the addition and multiplication of the cardinal 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 prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined. His argument is fundamental in the solution of the Halting problem and the proof of Gödel's first incompleteness theorem.

In 1895 and 1897, Cantor published a two-part paper in Mathematische Annalen under Felix Klein's editorship; these were his last significant papers on set theory.[37] 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 of well-ordered sets and ordinal numbers. Cantor attempts to prove that if A and B are sets with A equivalent to a subset of B and B equivalent to a subset of A, then A and B 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 name Cantor–Bernstein–Schroeder theorem.

#### One-to-one correspondence

A bijective function.

Cantor's 1874 Crelle paper was the first to invoke the notion of a 1-to-1 correspondence, though he did not use that phrase. He then began looking for a 1-to-1 correspondence between the points of the unit square and the points of a unit line segment. In an 1877 letter to Dedekind, Cantor proved a far stronger result: for any positive integer n, there exists a 1-to-1 correspondence between the points on the unit line segment and all of the points in an n-dimensional space. About this discovery Cantor famously wrote to Dedekind: "Je le vois, mais je ne le crois pas!" ("I see it, but I don't believe it!")[38] The result that he found so astonishing has implications for geometry and the notion of dimension.

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 from Jakob Steiner) or "equivalence" of sets: two sets are equivalent (have the same power) if there exists a 1-to-1 correspondence between them. Cantor defined countable sets (or denumerable sets) as sets which can be put into a 1-to-1 correspondence with the natural numbers, and proved that the rational numbers are denumerable. He also proved that n-dimensional Euclidean space Rn has the same power as the real numbers R, as does a countably infinite product of copies of R. While he made free use of countability as a concept, he did not write the word "countable" until 1883. Cantor also discussed his thinking about dimension, stressing that his mapping between the unit interval and the unit square was not a continuous one.

This paper, like the 1874 paper, displeased Kronecker, and Cantor wanted to withdraw it; however, Dedekind persuaded him not to do so and Weierstrass also supported its publication.[39] Nevertheless, Cantor never again submitted anything to Crelle.

#### Continuum hypothesis

Cantor was the first to formulate what later came to be known as the continuum 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 is exactly aleph-one, rather than just at least aleph-one). Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. His inability to prove the continuum hypothesis caused him considerable anxiety.[9]

The difficulty Cantor had in proving the continuum hypothesis has been underscored by later developments in the field of mathematics: a 1940 result by Gödel and a 1963 one by Paul Cohen together imply that the continuum hypothesis can neither be proved nor disproved using standard Zermelo–Fraenkel set theory plus the axiom of choice (the combination referred to as "ZFC").[40]

Discussions of set-theoretic paradoxes began to appear around the end of the nineteenth century. Some of these implied fundamental problems with Cantor's set theory program.[41] In an 1897 paper on an unrelated topic, Cesare Burali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.[11]

In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set A, the cardinal number of the power set of A is strictly larger than the cardinal number of A (this fact is now known as Cantor's theorem). This paradox, together with Burali-Forti's, led Cantor to formulate a concept called limitation of size,[42] according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as proper classes.

One common view among mathematicians is that these paradoxes, together with Russell's paradox, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naive set theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.[43]

## Philosophy, religion and Cantor's mathematics

The concept of the existence of an actual infinity was an important shared concern within the realms of mathematics, philosophy and religion. Preserving the orthodoxy 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.[44] He directly addressed this intersection between these disciplines in the introduction to his Grundlagen einer allgemeinen Mannigfaltigkeitslehre, where he stressed the connection between his view of the infinite and the philosophical one.[45] To Cantor, his mathematical views were intrinsically linked to their philosophical and theological implications—he identified the Absolute Infinite with God,[46] 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.[12]

Debate among mathematicians grew out of opposing views in the philosophy 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.[47] Mathematicians from three major schools of thought (constructivism and its two offshoots, intuitionism and finitism) opposed Cantor's theories in this matter. For constructivists such as Kronecker, this rejection of actual infinity stems from fundamental disagreement with the idea that nonconstructive proofs such as Cantor's diagonal argument are sufficient proof that something exists, holding instead that constructive 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.[6] 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.[48] Mathematicians such as Brouwer and especially Poincaré adopted an intuitionist stance against Cantor's work. Citing the paradoxes of set theory as an example of its fundamentally flawed nature, Poincaré held that "most of the ideas of Cantorian set theory should be banished from mathematics once and for all."[6] Finally, Wittgenstein's attacks were finitist: he believed that Cantor's diagonal argument conflated the intension of a set of cardinal or real numbers with its extension, thus conflating the concept of rules for generating a set with an actual set.[8]

Christian theologians saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the nature of God.[4] 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".[49] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:[50]

 “ ... 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.[51] ”

Cantor also believed that his theory of transfinite numbers ran counter to both materialism and determinism—and was shocked when he realized that he was the only faculty member at Halle who did not hold to deterministic philosophical beliefs.[52]

In 1888, Cantor published his correspondence with several philosophers on the philosophical implications of his set theory. In an extensive attempt to persuade Christian thinkers and authorities to adopt his views, Cantor had corresponded with Christian philosophers such as Tilman Pesch and Joseph Hontheim,[53] as well as theologians such as Cardinal Johannes Franzelin, who once replied by equating the theory of transfinite numbers with pantheism.[5] Cantor even sent one letter directly to Pope Leo XIII himself, and addressed several pamphlets to him.[50]

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 this metaphysical 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 famous assertion that "the essence of mathematics is its freedom."[54] These ideas parallel those of Edmund Husserl.[55]

Cantor's 1883 paper reveals that he was well aware of the opposition 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.[56] ”

Hence he devotes much space to justifying his earlier work, asserting that mathematical concepts may be freely introduced as long as they are free of contradiction and defined in terms of previously accepted concepts. He also cites Aristotle, Descartes, Berkeley, Leibniz, and Bolzano on infinity.

## Cantor's ancestry

Cantor was frequently described as Jewish in his lifetime.[57] Cantor's paternal grandparents were from Copenhagen, and fled to Russia from the disruption of the Napoleonic Wars. In his letters, Cantor referred to them as "Israelites". However, there is no direct evidence on whether his grandparents practiced Judaism; there is very little direct information on them of any kind.[58] Jakob Cantor, Cantor's grandfather, gave his children Christian saints' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they, or their ancestors, converted to Orthodox Christianity. Cantor's father, Georg Waldemar Cantor, was educated in the Lutheran mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. His mother, Maria Anna Böhm, was an Austrian born in Saint Petersburg and baptized Roman Catholic; she converted to Protestantism upon marriage. However, there is a letter from Cantor's brother Louis to their mother, saying

 “ "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..."[59] ”

which could be read to imply that she was of Jewish ancestry.[60]

Thus Cantor did not follow Judaism himself, but has nevertheless been called variously German, Jewish,[61] Russian, and Danish.

## Historiography

Until the 1970s, the chief academic publications on Cantor were two short monographs by 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 by Eric Temple Bell's Men of Mathematics (1937), which one of Cantor's modern biographers describes as "perhaps the most widely read modern book on the history of mathematics"; and as "one of the worst".[62] Bell presents Cantor's relationship with his father as Oedipal, 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, and fills the picture with stereotypes. 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.[63]

## Notes

1. ^ Grattan-Guinness 2000, p. 351
3. ^ Dauben 2004, p. 1.
4. ^ a b Dauben, 1977, p. 86; Dauben, 1979, pp. 120 & 143.
5. ^ a b Dauben, 1977, p. 102.
6. ^ a b c Dauben 1979, p. 266.
7. ^ Dauben 2004, p. 1. See also Dauben 1977, p. 89 15n.
8. ^ a b Rodych 2007
9. ^ a b c Dauben 1979, p. 280:"...the tradition made popular by Arthur Moritz Schönflies blamed Kronecker's persistent criticism and Cantor's inability to confirm his continuum hypothesis" for Cantor's recurring bouts of depression.
10. ^ a b 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's mental illness as "cyclic manic-depression".
11. ^ a b c Dauben 1979, p. 248
12. ^ a b Dauben 2004, pp. 8, 11 & 12-13.
13. ^ Hilbert 1926, p. 170; see Reid 1996, p. 177
14. ^ Dauben 1979, p. 163.
15. ^ Dauben 1979, p. 34.
16. ^ Dauben 1977, p. 89 15n.
17. ^ Dauben 1979, pp. 2–3; Grattan-Guinness 1971, pp. 354–355.
18. ^ Dauben 1979, p. 138.
19. ^ Dauben 1979, p. 139.
20. ^ a b Dauben 1979, p. 282.
21. ^ Dauben 1979, p. 136; Grattan-Guinness 1971, pp. 376–377. Letter dated June 21, 1884.
22. ^ Dauben 1979, pp. 281–283.
23. ^ Dauben 1979, p. 283.
24. ^ For a discussion of König's paper see Dauben 1979, 248–250. For Cantor's reaction, see Dauben 1979, p. 248; 283.
25. ^ Dauben 1979, p. 283–284.
26. ^ Dauben 1979, p. 284.
27. ^ a b Johnson 1972, p. 55.
28. ^ This paragraph is a highly abbreviated summary of the impact of Cantor's lifetime of work. More details and references can be found later.
29. ^ A countable 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".
30. ^ Reid 1996, p. 177.
31. ^ This follows closely the first part of Cantor's 1891 paper.
32. ^ Moore 1995, pp. 112 & 114; Dauben 2004, p. 1.
33. ^ For example, geometric problems posed by Galileo and John Duns Scotus suggested that all infinite sets were equinumerous — see Moore 1995, p. 114.
34. ^ For this, and more information on the mathematical importance of Cartan's work on set theory, see e.g., Suppes 1972.
35. ^ Education.fr
36. ^ Dauben 1977, p. 89.
37. ^ The English translation is Cantor 1955.
38. ^ Wallace 2003, p. 259.
39. ^ Dauben 1979, p. 69; 324 63n. The paper had been submitted in July 1877. Dedekind supported it, but delayed its publication due to Kronecker's opposition. Weierstrass actively supported it.
40. ^ 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 is W. Hugh Woodin. One of Gödel's last papers argues that the CH is false, and the continuum has cardinality Aleph-2.
41. ^ Dauben 1979, pp. 240–270; see especially pp. 241 & 259.
42. ^ Hallett 1986.
43. ^ Weir 1998, p. 766: "...it may well be seriously mistaken to think of Cantor's Mengenlehre [set theory] as naive..."
44. ^ Dauben 1979, p. 295.
45. ^ Dauben, 1979, p. 120.
46. ^ Hallett 1986, p. 13. Compare to the writings of Thomas Aquinas.
47. ^ Dauben 1979, p. 225
48. ^ Snapper 1979, p. 3
49. ^ Davenport 1997, p.3
50. ^ a b Dauben, 1977, p. 85.
51. ^ Cantor 1932, p. 404. Translation in Dauben 1977, p. 95.
52. ^ Dauben 1979, p. 296.
53. ^ Dauben, 1979, p. 144.
54. ^ Dauben 1977 pp. 91–93.
55. ^ On Cantor, Husserl, and Gottlob Frege, see Hill and Rosado Haddock (2000).
56. ^ Dauben 1979, p. 96.
57. ^ Jewish Encyclopedia, art. "Cantor, Georg"; Jewish Year Book 1896–1897, "List of Jewish Celebrities of the Nineteenth Century", p.119
58. ^ E.g., Grattan-Guinness's only evidence on the grandfather's date of death is that he signed papers at his son's engagement.
59. ^ Purkert and Ilgauds 1987, p. 15.
60. ^ 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 from Aczel 2000, pp. 93–94, from Louis' trip to Chicago in 1863. It is ambiguous in German, as in English, whether the recipient is included.
61. ^ Cantor was frequently described as Jewish in his lifetime. For example, Jewish Encyclopedia, art. "Cantor, Georg"; Jewish Year Book 1896–1897, "List of Jewish Celebrities of the Nineteenth Century", p.119; this list has a star against people with one Jewish parent, but Cantor is not starred.
62. ^ Grattan-Guinness 1971, p. 350.
63. ^ 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.)

## References

Older sources on Cantor's life should be treated with caution. See Historiography section above.
Primary literature in English
Primary literature in German
Secondary literature
• Aczel, Amir D. (2000). The mystery of the Aleph: Mathematics, the Kabbala, and the Human Mind. New York: Four Walls Eight Windows Publishing. ISBN 0760777780. A popular treatment of infinity, in which Cantor is frequently mentioned.
• Dauben, Joseph W. (1977). Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite. Journal of the History of Ideas 38.1.
• Dauben, Joseph W. (1979). Georg Cantor: his mathematics and philosophy of the infinite. Boston: Harvard University Press. The definitive biography to date. ISBN 978-0-691-02447-9
• Dauben, Joseph W. (1983). Georg Cantor and the Origins of Transfinite Set Theory. Scientific American 248.6:122-131
• Dauben, Joseph (1993, 2004). "Georg Cantor and the Battle for Transfinite Set Theory" in Proceedings of the 9th ACMS Conference (Westmont College, Santa Barbara, CA) (pp. 1–22). Internet version published in Journal of the ACMS 2004.
• Davenport, Anne A. (1997). The Catholics, the Cathars, and the Concept of Infinity in the Thirteenth Century. Isis 88.2:263–295.
• Grattan-Guinness, Ivor (1971). Towards a Biography of Georg Cantor. Annals of Science 27:345–391.
• Grattan-Guinness, Ivor (2000). The Search for Mathematical Roots: 1870–1940. Princeton University Press. ISBN 978-0691058580
• Hallett, Michael (1986). Cantorian Set Theory and Limitation of Size. New York: Oxford University Press. ISBN 0-19-853283-0
• Halmos, Paul (1998, 1960). Naive Set Theory. New York & Berlin: Springer. ISBN 3540900926
• Hill, C. O. & Rosado Haddock, G. E. (2000). Husserl or Frege? Meaning, Objectivity, and Mathematics. Chicago: Open Court. ISBN 0812695380 Three chapters and 18 index entries on Cantor.
• Johnson, Phillip E. (1972). The Genesis and Development of Set Theory. The Two-Year College Mathematics Journal 3.1:55–62.
• Meschkowski, Herbert (1983). Georg Cantor, Leben, Werk und Wirkung (George Cantor, Life, Work and Influence, in German). Wieveg, Braunschweig
• Moore, A.W. (1995, April). A brief history of infinity. Scientific American.4:112–116.
• Penrose, Roger (2004). The Road to Reality. Alfred A. Knopf. ISBN 0679776311 Chapter 16 illustrates how Cantorian thinking intrigues a leading contemporary theoretical physicist.
• Purkert, Walter & Ilgauds, Hans Joachim (1985). Georg Cantor: 1845–1918. Birkhäuser. ISBN 0-8176-1770-1
• Reid, Constance (1996). Hilbert. New York: Springer-Verlag. ISBN 0387049991
• Rucker, Rudy (2005, 1982). Infinity and the Mind. Princeton University Press. ISBN 0553255312 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.
• Snapper, Ernst (1979). The Three Crises in Mathematics: Logicism, Intuitionism and Formalism. Mathematics Magazine 524:207–216.
• Suppes, Patrick (1972, 1960). Axiomatic Set Theory. New York: Dover. ISBN 0486616304 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.
• Wallace, David Foster (2003). Everything and More: A Compact History of Infinity. New York: W.W. Norton and Company. ISBN 0393003388
• Weir, Alan (1998). Naive Set Theory is Innocent!. Mind 107.428:763–798.

# Quotes

Up to date as of January 14, 2010

### From Wikiquote

In mathematics the art of asking questions is more valuable than solving problems.

Georg Ferdinand Ludwig Philipp Cantor () was a Russian-born mathematician and philosopher of Danish and Austrian descent, best known as the creator of set theory, and of Cantor's theorem which implies the existence of an "infinity of infinities."

## Sourced

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.
Each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.
The least particle ought to be considered as a world full of an infinity of different creatures.
• In re mathematica ars proponendi pluris facienda est quam solvendi.
• In mathematics the art of asking questions is more valuable than solving problems.
• Doctoral thesis (1867); variant translation: In mathematics the art of proposing a question must be held of higher value than solving it.
• 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.
• Grundlagen einer allgemeinen Mannigfaltigkeitslehre [Foundations of a General Theory of Aggregates] (1883)
• Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!
• Letter (1885), written after Gösta Mittag-Leffler persuaded him to withdraw a submission to Mittag-Leffler's journal Acta Mathematica, telling him it was "about one hundred years too soon."
• The old and oft-repeated proposition "Totum est majus sua parte" [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts "totum" and "pars". Unfortunately, however, this "axiom" is used innumerably often without any basis and in neglect of the necessary distinction between "reality" and "quantity", on the one hand, and "number" and "set", on the other, precisely in the sense in which it is generally false.
• "Über unendliche, lineare Punktmannigfaltigkeiten" in Mathematische Annalen 20 (1882) Quoted in "Cantor's Grundlagen and the paradoxes of Set Theory" by William W. Tait
• There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.
• "Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"] - Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886)
• In order for there to be a variable quantity in some mathematical study, the domain of its variability must strictly speaking be known beforehand through a definition. However, this domain cannot itself be something variable, since otherwise each fixed support for the study would collapse. Thus this domain is a definite, actually infinite set of values. Hence each potential infinite, if it is rigorously applicable mathematically, presupposes an actual infinite.
• "Über die verschiedenen Ansichten in Bezug auf die actualunendlichen Zahlen" ["Over the different views with regard to the actual infinite numbers"] - Bihand Till Koniglen Svenska Vetenskaps Akademiens Handigar (1886)
• The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.
• "Mitteilungen" (1887-8)
• Er ist aber in Kopenhagen geboren, von israelitischen Eltern, die der dortigen portugisischen Judengemeinde.
• However, he was born in Copenhagen, of Jewish parents, of the Portuguese Jewish community there.
• Of his father. In a letter written by Georg Cantor to Paul Tannery in 1896 (Paul Tannery, Memoires Scientifique 13 Correspondance, Gauthier-Villars, Paris, 1934, p. 306)
• The totality of all alephs cannot be conceived as a determinate, well-defined, and also a finished set. This is the punctum saliens, and I venture to say that this completely certain theorem, provable rigorously from the definition of the totality of all alephs, is the most important and noblest theorem of set theory. One must only understand the expression "finished" correctly. I say of a set that it can be thought of as finished (and call such a set, if it contains infinitely many elements, "transfinite" or "suprafinite") if it is possible without contradiction (as can be done with finite sets) to think of all its elements as existing together, and to to think of the set itself as a compounded thing for itself; or (in other words) if it is possible to imagine the set as actually existing with the totality of its elements.
• Every transfinite consistent multiplicity, that is, every transfinite set, must have a definite aleph as its cardinal number.
• Letter to Richard Dedekind (1899), as translated in From Frege to Gödel : A Source Book in Mathematical Logic, 1879-1931 (1967) by Jean Van Heijenoort, p. 117
• I have never proceeded from any Genus supremum of the actual infinite. Quite the contrary, I have rigorously proved that there is absolutely no Genus supremum of the actual infinite. What surpasses all that is finite and transfinite is no Genus; it is the single, completely individual unity in which everything is included, which includes the Absolute, incomprehensible to the human understanding. This is the Actus Purissimus, which by many is called God.
I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not — I do not say divisible — but actually divisible; and consequently the least particle ought to be considered as a world full of an infinity of different creatures.
• As quoted in Out of the Mouths of Mathematicians : A Quotation Book for Philomaths (1993) by Rosemary Schmalz.
• A set is a Many that allows itself to be thought of as a One.
• As quoted in Infinity and the Mind (1995) by Rudy Rucker. ~ ISBN 0691001723
The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.
• The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.
• As quoted in Infinity and the Mind (1995) by Rudy Rucker.
• The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.
• As quoted in Mind Tools: The Five Levels of Mathematical Reality (1988) by Rudy Rucker. ~ ISBN 0395468108
• That from the outset they expect or even impose all the properties of finite numbers upon the numbers in question, while on the other hand the infinite numbers, if they are to be considered in any form at all, must (in their contrast to the finite numbers) constitute an entirely new kind of number, whose nature is entirely dependent upon the nature of things and is an object of research, but not of our arbitrariness or prejudices.
• Letter to Gustac Enestrom, as quoted in Georg Cantor : His Mathematics and Philosophy of the Infinite (1990) by Joseph Warren Dauben ~ ISBN 0691024472
• This view [of the infinite], which I consider to be the sole correct one, is held by only a few. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last!
• As quoted in Journey Through Genius (1990) by William Dunham ~ ISBN 0471500305
• My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.
• As quoted in Journey Through Genius (1990) by William Dunham
• What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending ladder of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.
• As quoted in Understanding the Infinite (1994) by Shaughan Lavine ~ ISBN 0674921178
• The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are like each other in their innermost being; for the former like the latter are definite delimited forms or modifications of the actual infinite.
• As quoted in Understanding the Infinite (1994) by Shaughan Lavine
• I entertain no doubts as to the truths of the transfinites, which I recognized with God"s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.
• As quoted in Modern Mathematicians, (1995) by Harry Henderson. ~ ISBN 0816032351

### From Kant to Hilbert (1996)

Quotes of Cantor in From Kant to Hilbert : A Source Book in the Foundations of Mathematics (1996) by William Bragg Ewald ~ ISBN 0198505361
• As for the mathematical infinite, to the extent that it has found a justified application in science and contributed to is usefulness, it seems to me that it has hitherto appeared principally in the role of a variable quantity , which either grows beyond all bounds or diminishes to any desired minuteness, but always remains finite. I call this the improper infinite [das Uneigentlich-unendliche].
• Infinity, in its first form (the improper-infinite) presents itself as a variable finite [veranderliches Endliches]; in the other form (which I call the proper infinite [Eigentlich-unendliche]) it appears as a thoroughly determinate [bestimmtes] infinite.
• What I declare and believe to have demonstrated in this work as well as in earlier papers is that following the finite there is a transfinite (transfinitum)--which might also be called supra-finite (suprafinitum), that is, there is an unlimited ascending ladder of modes, which in its nature is not finite but infinite, but which can be determined as can the finite by determinate, well-defined and distinguishable numbers.
• Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.
• Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established. In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished. As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real.
• The essence of mathematics lies entirely in its freedom.
• Variant translation: The essence of mathematics is in its freedom.
• If there is some determinate succession of defined whole real numbers, among which there exists no greatest, on the basis of this second principle of generation a new number is obtained which is regarded as the limit of those numbers, i.e. is defined as the next greater number than all of them.

• No one shall expel us from the Paradise that Cantor has created.
• David Hilbert, in "Über das Unendliche" [On the Infinite], in Mathematische Annalen 95 (1925)
• Variant translation: No one shall expel us from the Paradise that Cantor has created for us.
• Why was Cantor so vehemently opposed to infinitesimals? In his valuable essay, "The Metaphysics of the Calculus," Abraham Robinson suggests that Cantor already had enough problems trying to defend transfinite numbers. It seems likely that, consciously or otherwise, Cantor deemed it politically wise to go along with orthodox mathematicians on the question of infinitesimals. Cantor's stance might be compared to that of a pro-marijuana Congressional candidate who advocates harsh penalties for the sale or use of heroin.
• After being relegated to an obscure mid-tier university, blocked from leading journals and openly mocked by his peers, including his former mentor, the late 19th century German mathematician found refuge for his groundbreaking work on infinities in, of all places, the Roman Catholic Church... Catholic theologians welcomed Cantor's ideas, which provided a workable way of understanding mathematical infinities, as evidence that humans could grasp the infinite and could also, therefore, have a greater understanding of God, himself infinite.
What a welcome relief this must have been to the chronically depressed Cantor! As John D. Barrow writes in The Infinite Book: A Short Guide to the Boundless, Timeless and Endless, Cantor "started to tell his friends that he had not been the inventor of the ideas about infinity that he had published. He was merely a mouthpiece, inspired by God to communicate parts of the mind of God to everyone else."
• I discovered the works of Euler and my perception of the nature of mathematics underwent a dramatic transformation. I was de-Bourbakized, stopped believing in sets, and was expelled from the Cantorian paradise.

# Wikibooks

Up to date as of January 23, 2010

### From Wikibooks, the open-content textbooks collection

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# Simple English

Georg Cantor was an important German mathematician.