**Georg Cantor** was a Russian-born mathematician who can be considered as the founder of set theory and introduced the concept of infinite numbers with his discovery of cardinal numbers. He also advanced the study of trigonometric series.

- Georg Waldemar Cantor was born in Denmark and he was a man with a deep love of culture and the arts.
- Certainly Georg inherited considerable musical and artistic talents from his parents being an outstanding violinist.
- At first they lived in Wiesbaden, where Cantor attended the Gymnasium, then they moved to Frankfurt.
- Cantor studied at the Realschule in Darmstadt where he lived as a boarder.
- Cantor moved to the University of Berlin where he became friends with Hermann Schwarz who was a fellow student.
- Cantor attended lectures by Weierstrass, Kummer and Kronecker.
- While at Berlin Cantor became much involved with a student Mathematical Society, being president of the Society during 1864-65.
- After receiving his doctorate in 1867, Cantor taught at a girl's school in Berlin.
- At Halle the direction of Cantor's research turned away from number theory and towards analysis.
- This was due to Heine, one of his senior colleagues at Halle, who challenged Cantor to prove the open problem on the uniqueness of representation of a function as a trigonometric series.
- Cantor solved the problem proving uniqueness of the representation by April 1870.
- Cantor was promoted to Extraordinary Professor at Halle in 1872 and in that year he began a friendship with Dedekind whom he had met while on holiday in Switzerland.
- Cantor published a paper on trigonometric series in 1872 in which he defined irrational numbers in terms of convergent sequences of rational numbers.
- Dedekind published his definition of the real numbers by "Dedekind cuts" also in 1872 and in this paper Dedekind refers to Cantor's 1872 paper which Cantor had sent him.
- In 1873 Cantor proved the rational numbers countable, i.e. they may be placed in one-one correspondence with the natural numbers.
- Twenty years later, in this 1874 work, Cantor showed that in a certain sense 'almost all' numbers are transcendental by proving that the real numbers were not countable while he had proved that the algebraic numbers were countable.
- Cantor pressed forward, exchanging letters throughout with Dedekind.
- The year 1874 was an important one in Cantor's personal life.
- A major paper on dimension which Cantor submitted to Crelle's Journal in 1877 was treated with suspicion by Kronecker, and only published after Dedekind intervened on Cantor's behalf.
- Cantor greatly resented Kronecker's opposition to his work and never submitted any further papers to Crelle's Journal.
- Between 1879 and 1884 Cantor published a series of six papers in Mathematische Annalen designed to provide a basic introduction to set theory.
- However there were a number of problems which occurred during these years which proved difficult for Cantor.
- Although he had been promoted to a full professor in 1879 on Heine's recommendation, Cantor had been hoping for a chair at a more prestigious university.
- His long standing correspondence with Schwarz ended in 1880 as opposition to Cantor's ideas continued to grow and Schwarz no longer supported the direction that Cantor's work was going.
- Cantor drew up a list of three mathematicians to fill Heine's chair and the list was approved.
- It was certainly a severe blow to Cantor when Dedekind declined the offer in the early 1882, and the blow was only made worse by Heinrich Weber and then Mertens declining too.
- After a new list had been drawn up, Wangerin was appointed but he never formed a close relationship with Cantor.
- The rich mathematical correspondence between Cantor and Dedekind ended later in 1882.
- Almost the same time as the Cantor-Dedekind correspondence ended, Cantor began another important correspondence with Mittag-Leffler.
- Soon Cantor was publishing in Mittag-Leffler's journal Acta Mathematica but his important series of six papers in Mathematische Annalen also continued to appear.
- Firstly Cantor realised that his theory of sets was not finding the acceptance that he had hoped and the Grundlagen was designed to reply to the criticisms.
- At the end of May 1884 Cantor had the first recorded attack of depression.
- Mathematical worries began to trouble Cantor at this time, in particular he began to worry that he could not prove the continuum hypothesis, namely that the order of infinity of the real numbers was the next after that of the natural numbers.
- All was not going well in other ways too, for in 1885 Mittag-Leffler persuaded Cantor to withdraw one of his papers from Acta Mathematica when it had reached the proof stage because he thought it "...
- Mittag-Leffler meant this as a kindness but it does show a lack of appreciation of the importance of Cantor's work.
- The correspondence between Mittag-Leffler and Cantor all but stopped shortly after this event and the flood of new ideas which had led to Cantor's rapid development of set theory over about 12 years seems to have almost stopped.
- In 1886 Cantor bought a fine new house on Händelstrasse, a street named after the German composer Handel.
- Cantor chaired the first meeting of the Association in Halle in September 1891, and despite the bitter antagonism between himself and Kronecker, Cantor invited Kronecker to address the first meeting.
- Cantor was elected president of the Deutsche Mathematiker-Vereinigung at the first meeting and held this post until 1893.
- Cantor published a rather strange paper in 1894 which listed the way that all even numbers up to 1000 could be written as the sum of two primes.
- Since a verification of Goldbach's conjecture up to 10000 had been done 40 years before, it is likely that this strange paper says more about Cantor's state of mind than it does about Goldbach's conjecture.
- The rather long gap between the two papers is due to the fact that although Cantor finished writing the second part six months after the first part was published, he hoped to include a proof of the continuum hypothesis in the second part.
- In 1897 Cantor attended the first International Congress of Mathematicians in Zürich.
- Hurwitz openly expressed his great admiration of Cantor and proclaimed him as one by whom the theory of functions has been enriched.
- At the Congress Cantor met Dedekind and they renewed their friendship.
- By the time of the Congress, however, Cantor had discovered the first of the paradoxes in the theory of sets.
- Cantor began a correspondence with Dedekind to try to understand how to solve the problems but recurring bouts of his mental illness forced him to stop writing to Dedekind in 1899.
- Whenever Cantor suffered from periods of depression he tended to turn away from mathematics and turn towards philosophy and his big literary interest which was a belief that Francis Bacon wrote Shakespeare's plays.
- In October 1899 Cantor applied for, and was granted, leave from teaching for the winter semester of 1899-1900.
- Cantor also spent some time in sanatoria, at the times of the worst attacks of his mental illness, from 1899 onwards.
- In 1905 Cantor wrote a religious work after returning home from a spell in hospital.
- Cantor had hoped to meet with Russell who had just published the Principia Mathematica.
- The following year Cantor was awarded the honorary degree of Doctor of Laws by the University of St Andrews but he was too ill to receive the degree in person.
- Cantor retired in 1913 and spent his final years ill with little food because of the war conditions in Germany.
- A major event planned in Halle to mark Cantor's 70th birthday in 1915 had to be cancelled because of the war, but a smaller event was held in his home.

Born 3 March 1845, St Petersburg, Russia. Died 6 January 1918, Halle, Germany.

View full biography at MacTutor

Analysis, Ancient Indian, Geometry, Origin Russia, Set Theory, Special Numbers And Numerals, Topology

Axioms: 1

Chapters: 2 3

Definitions: 4 5 6

Motivations: 7 8 9

Parts: 10 11 12 13

Proofs: 14

Propositions: 15 16

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive