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He was born in Saint 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.
Cantor recognized that infinite setsInfinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. In theology, for instance in the work of Duns Scotus, the infinity of God carries the sense not so much of quantity (leading to the question can have different sizes, distinguished between countable and uncountable setsThis article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now and proved that the set of all rational numbersIn mathematics, a rational number (or informally fraction is a ratio of two integers, usually written as the vulgar fraction a ''b where b is not zero. The set of all rational numbers is denoted by Q or in blackboard bold. Using the set-builder notation i Q is countable while the set of all real numbersIn mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to " imaginary number". Real numbers may 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 argumentSet theory Note: in order to fully understand this article you may want to refer to the set theory portion of the table of mathematical symbols. Cantor's diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not cou. In his later years, he tried in vain to prove the continuum hypothesisIn mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than t. 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)