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.
Works
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.
