Cantor, Georg

Cantor, Georg (gā`ôrkh kän`tôr), 1845–1918, German mathematician, b. St. Petersburg. He studied under Karl Weierstrass and taught (1869–1913) at the Univ. of Halle. He is known for his work on transfinite numbers and on the development of set theory, which is the basis of modern analysis, as well as for his definition of irrational numbers. His approach to the concept of the infinite revolutionized mathematics by challenging the processes of deductive reasoning and led to a critical investigation of the foundations of mathematics.

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Cantor, Georg

(born March 3, 1845, St. Petersburg, Russia—died Jan. 6, 1918, Halle, Ger.) German mathematician, founder of set theory. He was the first to examine number systems, such as the rational numbers and the real numbers, systematically as complete entities, or sets. This led him to the surprising discovery that not all infinite sets are the same size. In particular, he showed that the rational numbers could be put in a one-to-one correspondence with the counting numbers; hence the set is countable. He also showed that no such correspondence is possible for the much larger set of irrational numbers; hence they are known as an uncountable set. His investigations led him to the classification of transfinite numbers, which are, informally speaking, degrees of infinity.

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Cantor, Georg 

Born Mar. 3, 1845, in St. Petersburg; died Jan. 6, 1918, in Halle. German mathematician.

Cantor graduated from the University of Berlin in 1867. He developed the theory of infinite sets and the theory of transfinite numbers. In 1874 he proved the uncountability of the set of all real numbers, thus establishing the existence of inequivalent (that is, having different powers) infinite sets; he formulated (1878) the general concept of the power of a set. Between 1879 and 1884, Cantor systematically set forth the principles of his study of infinity. He introduced the concepts of limit point and derived set, constructed an example of a perfect set, developed one of the theories of irrational numbers, and formulated one of the axioms of continuity. In 1897 he retired from scientific work. Cantor’s ideas encountered intense opposition from his contemporaries, in particular from L. Kronecker, but they subsequently exerted great influence on the development of mathematics.

WORKS

Gesammelte Abhandlungen mathematischen und philosophischen Inhalts Berlin, 1932.
In Russian translation:
“Uchenie o mnozhestvakh.” In the collection Novye idei v matematike, no. 6. St. Petersburg, 1914.