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Georg Cantor


Russian-born German mathematician best known as the creator of set theory

Georg Ferdinand Ludwig Philipp Cantor (St. Petersburg , 3 March 1845 - Halle , 6 January 1918) was a mathematician German, father of modern set theory . Cantor's set theory has expanded to include within it the concept of transfinite numbers , cardinal numbers and ordinal.

Cantor was born in St. Petersburg, son of Georg Cantor Woldemar, a stock trader Danish , and Marie Anna Böhm, a musician born in Russia, but of Austrian origin. In 1856 due to the state of health of the father, the family moved to Germany and Georg continued his education at German schools, first in Darmstadt, then in Switzerland ETH Zurich, finally earning his doctorate at the ' University of Berlin in 1867 with a thesis on the theory of numbers : De aequationibus Secundi gradus indeterminatis.

Cantor acknowledged that the infinite sets can have different cardinalities, separated the sets in countable and uncountable more than and proved that the set of all rational numbers Q is countable while the set of all real numbers R is more than countable, demonstrating so that there are at least two orders of infinity. He also invented the symbol that is used today to indicate the actual numbers. The method which he used to conduct his demonstration is known as the method of the diagonal Cantor. Later, he tried in vain to prove the ' continuum hypothesis . Cantor formulated a very important principle for the definition of real numbers, called the principle of localization , which is fundamental in order to operate on the above number field.

During the second half of his life he suffered from bouts of depression , which seriously compromised his ability as a mathematician and forced him to repeated hospitalizations. He began to read texts of literature and religion, in which he developed his concept of ' infinite absolute who identified with God . He wrote:

"The actual infinity comes in three contexts: first when it is realized in a more complete form, in essence mystical completely independent in God , which I call Absolute Infinite or simply Absolute; secondly when making in the contingent world, created, third when the mind grasps it in abstracto as a magnitude, number, or a type of mathematical order. "

Impoveritosi during the First World War , died in Halle where he was hospitalized in a psychiatric hospital. Leopold Kronecker judged his findings "meaningless".

A Cantor has been named the Cantor crater on the Moon .

Cantor gave rise to the theory of sets ( 1874 - 1884 ) was the first to understand that infinite sets can have different sizes: first showed that given any set , there is the set of all possible subsets , called the ' power set of . Then demonstrated that the power set of an infinite set has a magnitude greater than the magnitude of the same (this fact is now known by the name of theorem Cantor ). So there is an infinite hierarchy of sizes of infinite sets, from which arise the numbers cardinal and ordinal transfinite, and their peculiar arithmetic. To denote the cardinal numbers used the letter of ' Hebrew alphabet aleph with a natural number as an index ( Alef zero), for the ordinals used the letter of ' greek alphabet omega.

The innovative Cantorian theory, opposed during the life of its creator, has been fully accepted by modern mathematicians, who recognized in the theory of transfinite sets one paradigm shift of the first magnitude.


Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Numbers . Journal für die reine und Mathematik angewandte . 1874. (First written on set theory).
Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen , in: Mathematische Annalen 5 (1872) 123-132.
Über die verschiedenen Standpunkte in Bezug auf das aktuale Unendliche , 1886.
Über eine der elementary Frage Mannigfaltigkeitslehre , 1890/91.
Georg Cantor: Beiträge zur Begründung der transfiniten Mengenlehre (1. Artikel), in: Mathematische Annalen 46 (1895), S. 481-512.
Georg Cantor: Beiträge zur Begründung der transfiniten Mengenlehre (2. Artikel), in: Mathematische Annalen 49 (1897), S. 207-246.
Georg Cantor: Gesammelte Abhandlungen Mathematischen und philosophischen Inhalts. Hrsg. st. E. Zermelo. Berlin: Springer, 1932. (Reprint: Springer, 1980.)
Georg Cantor: Briefe. Hrsg. st. H. Meschkowski. Berlin: Springer 1991.

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