This caused it to encounter 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; see Controversy over Cantor's theory.

Writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory", which he dismissed as "utter nonsense" that is "laughable" and "wrong".

[10] Cantor's recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile attitude of many of his contemporaries,[11] though some have explained these episodes as probable manifestations of a bipolar disorder.

After receiving a substantial inheritance upon his father's death in June 1863,[19] Cantor transferred to the University of Berlin, attending lectures by Leopold Kronecker, Karl Weierstrass and Ernst Kummer.

During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met at Interlaken in Switzerland two years earlier while on holiday.

[23] Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague,[24] perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.

He also began an intense study of Elizabethan literature, thinking there might be evidence that Francis Bacon wrote the plays attributed to William Shakespeare (see Shakespearean authorship question); this ultimately resulted in two pamphlets, published in 1896 and 1897.

[35] Although Ernst Zermelo demonstrated less than a day later that König's proof had failed, Cantor remained shaken, and momentarily questioning God.

The following year, St. Andrews awarded Cantor an honorary doctorate, but illness precluded his receiving the degree in person.

The Continuum hypothesis, introduced by Cantor, was presented by David Hilbert as the first of his twenty-three open problems in his address at the 1900 International Congress of Mathematicians in Paris.

While extending the notion of number by means of his revolutionary concept of infinite cardinality, Cantor was paradoxically opposed to theories of infinitesimals of his contemporaries Otto Stolz and Paul du Bois-Reymond, describing them as both "an abomination" and "a cholera bacillus of mathematics".

[45] The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 paper,[38] "Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").

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 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.

[70] In 1923, John von Neumann developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor's—namely, by identifying collections that are not sets and treating them differently.

[74] 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.

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.

[83] Cantor strongly believed that this view was a misinterpretation of infinity, and was convinced that set theory could help correct this mistake:[84] "... 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.".

[85] Prominent neo-scholastic German philosopher Constantin Gutberlet was in favor of such theory, holding that it didn't oppose the nature of God.

[8] Cantor also believed that his theory of transfinite numbers ran counter to 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.

[84] 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.

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.

[96] Cantor's mother, Maria Anna Böhm, was an Austro-Hungarian 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, 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 ...[96]("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.

[98] In a 1971 article entitled "Towards a Biography of Georg Cantor", the British historian of mathematics Ivor Grattan-Guinness mentions (Annals of Science 27, pp.

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.

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 ..."[96]Until the 1970s, the chief academic publications on Cantor were two short monographs by Arthur Moritz Schönflies (1927) – largely the correspondence with Mittag-Leffler – and Fraenkel (1930).

There are other legends, independent of Bell – including one that labels Cantor's father a foundling, shipped to Saint Petersburg by unknown parents.

[106] Writes Dauben: Cantor devoted some of his most vituperative correspondence, as well as a portion of the Beiträge, to attacking what he described at one point as the 'infinitesimal Cholera bacillus of mathematics', which had spread from Germany through the work of Thomae, du Bois Reymond and Stolz, to infect Italian mathematics ... Any acceptance of infinitesimals necessarily meant that his own theory of number was incomplete.