Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a German mathematician who 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).
He was born in Saint Petersburg Russia, the son of a Danish merchant, George 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.
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. By 1897, he had discovered several paradoxes in elementary set theory.
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.
Cantor's innovative mathematics faced significant resistance, especially by Leopold Kronecker, Hermann Weyl, L.E.J. Brouwer, Henri Poincaré and Ludwig Wittgenstein. The vast majority of working mathematicians accept Cantor's work on transfinite sets and recognize it as a paradigm shift of major importance. (See intuitionism and infinity)
"No one shall expel us from the Paradise that Cantor has created." David Hilbert