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| He recognized that ] can have different sizes, distinguished between ] and ] ] and proved that the set of all ] '''Q''' is countable while the set of all ] '''R''' is uncountable and hence strictly bigger. The proof uses his celebrated ]. In his later years, he tried in vain to prove the ]. By ], he had discovered several ] in elementary set theory. | He recognized that ] can have different sizes, distinguished between ] and ] ] and proved that the set of all ] '''Q''' is countable while the set of all ] '''R''' is uncountable and hence strictly bigger. The proof uses his celebrated ]. In his later years, he tried in vain to prove the ]. By ], he had discovered several ] in elementary set theory. | ||
| Throughout the second half of his life he suffered from bouts of ], which severely affected his ability to work and forced him to become hospitalized repeatedly. He started to publish about ] and ], and developed his concept of the ] which he equated with ]. He was impoverished during ] and died in a ] in ]. | Throughout the second half of his life he suffered from bouts of ], which severely affected his ability to work and forced him to become hospitalized repeatedly. The discovery of ] led to a ] from which he never recovered. He started to publish about ] and ], and developed his concept of the ] which he equated with ]. He was impoverished during ] and died ] in a ] in ]. | ||
| Cantor's innovative mathematics faced significant resistance during his lifetime. Modern mathematics completely accepts Cantor's work on transfinite sets and recognizes it as a paradigm shift of major importance. | Cantor's innovative mathematics faced significant resistance during his lifetime. Modern mathematics completely accepts Cantor's work on transfinite sets and recognizes it as a paradigm shift of major importance. | ||
Revision as of 02:45, 27 November 2002
Georg Ferdinand Ludwig Philipp Cantor was a German mathematician, born: March 3 1845, Saint Petersburg Russia, died: January 6 1918, Halle, Germany. He is best known for having created set theory and the concept of transfinite numbers, including the cardinal and ordinal number classes.
He 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 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.
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. The discovery of Russell's paradox led to a nervous breakdown from which he never recovered. He started to publish about literature and religion, and developed his concept of the Absolute Infinite which he equated with god. He was impoverished during World War I and died insane in a sanatorium in 1918.
Cantor's innovative mathematics faced significant resistance during his lifetime. Modern mathematics completely accepts Cantor's work on transfinite sets and recognizes it as a paradigm shift of major importance.
See also:
External link:
- http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Cantor.html Cantor's biography.