Cantor introduced fundamental constructions in set theory, such as the of a set , which is the set of all possible of . He later proved that the size of the power set of is strictly larger than the size of , even when is an infinite set; this result soon became known as . Cantor developed an entire theory and , called and , which extended the arithmetic of the natural numbers. His notation for the cardinal numbers was the Hebrew letter () with a natural number subscript; for the ordinals he employed the Greek letter ω (). This notation is still in use today.
"Very little is known for sure about the origin and education of George Woldemar Cantor." However, Cantor was frequently described as Jewish in his lifetime. Cantor's paternal grandparents were from , and fled to Russia from the disruption of the . There is very little direct information on his grandparents, consequently their level of Jewish observance is unknown. Jakob Cantor, Cantor's grandfather, gave his children ' names. Further, several of his grandmother's relatives were in the Czarist civil service, which would not welcome Jews, unless they, or their ancestors, to Orthodox Christianity. Cantor's father, Georg Waldemar Cantor, was educated in the mission in Saint Petersburg, and his correspondence with his son shows both of them as devout Lutherans. His mother, Maria Anna Böhm, was an born in Saint Petersburg and baptized ; she converted to upon marriage. However, there is a letter from Cantor's brother Louis to their mother, saying
The German Georg Cantor was an outstanding violinist, ..
showed that CH cannot be disproved from ZF, even if the (AC) is adopted (making ZFC). Gödel's proof shows that CH and AC both hold in the L, an of ZF set theory, assuming only the axioms of ZF. The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are with ZF, provided ZF itself is consistent. The latter condition cannot be proved in ZF itself, due to , but is widely believed to be true and can be proved in stronger set theories.
The Continuum Hypothesis | judgybitch
Cantor was promoted to Extraordinary Professor in 1872 and made full Professor in 1879. To attain the latter rank at the age of 34 was a notable accomplishment, but Cantor desired a at a more prestigious university, in particular at Berlin, at that time the leading German university. However, his work encountered too much opposition for that to be possible. Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians. Worse yet, Kronecker, a well-established figure within the mathematical community and Cantor's former professor, fundamentally disagreed with the thrust of Cantor's work. Kronecker, now seen as one of the founders of the , disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties. Cantor came to believe that Kronecker's stance would make it impossible for Cantor to ever leave Halle.
what is the continuum hypothesis
With infinite sets such as the set of or , this becomes morecomplicated to demonstrate. The rational numbers seemingly form acounterexample to the continuum hypothesis: the rationals form aproper superset of the integers, and a proper subset of the reals,so intuitively, there are more rational numbers than integers, andfewer rational numbers than real numbers. However, this intuitiveanalysis does not take account of the fact that all three sets are. It turnsout the rational numbers can actually be placed in one-to-onecorrespondence with the integers, and therefore the set of rationalnumbers is the same size (cardinality) as the set ofintegers: they are both .
Continuum Hypothesis | The Book of Threes
Cantor gave two proofs that the cardinality of the set of is strictly smallerthan that of the set of ; the second of these is his . Hisproofs, however, give no indication of the extent to which thecardinality of the natural numbers is less than that of the realnumbers. Cantor proposed the continuum hypothesis as a possiblesolution to this question.