# Georg Cantor

**Georg Ferdinand Ludwig Philipp Cantor** (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. He is best known as the creator of modern set theory. He is recognized by mathematicians for having extended set theory to the concept of transfinite numbers, including the cardinal and ordinal number classes. Cantor is also known for his work on the unique representations of functions by means of trigonometric series (a generalized version of a Fourier series).

Cantor's innovative mathematics faced significant resistance, especially by Leopold Kronecker, Hermann Weyl, L.E.J. Brouwer, Henri Poincaré and Ludwig Wittgenstein. The hostile attitude of many contemporaries is believed to have severely aggravated Cantor's emotional ailments and to have caused several nervous breakdowns.

Today, the vast majority of mathematicians accept Cantor's work on transfinite sets and recognize it as a paradigm shift of major importance. In the words of David Hilbert: "No one shall expel us from the Paradise that Cantor has created."

## Biography

Cantor was born in St Petersburg, Russia, the son of a Danish merchant, Georg Waldemar Cantor, and a Russian musician, Maria Anna Böhm. In 1856 the family moved to Germany and he continued his education in German schools, earning his doctorate from the University of Berlin in 1867.

In 1890 he founded together with other mathematicians the Deutsche Mathematiker-Vereinigung and became the first president of the society.

Cantor recognized that infinite sets can have different sizes, distinguished between countable and uncountable sets and proved that the set of all rational numbers **Q** is countable while the set of all real numbers **R** is uncountable and hence strictly bigger. The original proof of this, devised in December 1873 and published in early 1874, used a moderately complicated reduction argument in which one starts with a countable list of real numbers and an interval on the real line. Then, one takes the first two elements from the list that are in the interval, and forms an interval from that. Exhausting onward, we find that there exists an element that is not in the list. His later 1891 proof uses his celebrated diagonal argument. In his later years, he tried in vain to prove the continuum hypothesis.

He also invented the symbol today used to represent all real numbers.

Throughout the second half of his life he suffered from bouts of depression, which severely affected his ability to work and forced him to become hospitalized repeatedly. This recurrent depression would probably be diagnosed as bipolar disorder today. Indeed, one can easily see this degeneration in his publication of a verification of Goldbach's conjecture for all integers less than 1000 (a verification up to 10000 had been published decades before). He started to publish about literature, attempting to prove that Francis Bacon was the true author of Shakespeare's works, and religion in which he developed his concept of the Absolute Infinite which he equated with God. He was impoverished during World War I and died in a mental hospital in Halle, Germany.

## See also

- Cantor dust
- Cantor set
- Cantor's theorem
- Heine-Cantor theorem
- Intuitionism
- Infinity
- Aleph number
- Cantor's back-and-forth method
- Cantor medal - award by the Deutsche Mathematiker-Vereinigung in honor of Georg Cantor
- Controversy over Cantor's Theory

## External links

- O'Connor, John J., and Edmund F. Robertson. "Georg Cantor".
*MacTutor History of Mathematics archive*. - Selections from Cantor's philosophical writing
- Text of the 1891 diagonal proof

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