Cantor's paradox

Georg Cantor described in the years 1897-1899 several antinomies, by which he proved that certain classes are not sets. Prove his evidence that he did not naive - contradictory concept of a set, which is often claimed because of Cantor's set definition. He separated already amounts as consistent multiplicities of inconsistent multiplicities that are now called proper classes. Because Cantor his antinomies not published, but only by letter told to David Hilbert and Richard Dedekind, his set theory was often judged wrongly as naive set theory. It was not until the publication of his letters in 1932 by Zermelo made ​​known to the inventor of set theory, the antinomy problem was aware very early on.

First Cantor's antinomy

1897 showed Cantor that the class of all ( transfinite ) cardinal numbers, the " totality of Aleph ", not a lot, but a real class, through an indirect proof: If this totality a lot, it would be a greater Aleph, which as an element would belong to this totality and would not include.

The first Cantor's antinomy is to be distinguished from Burali - Forti paradox from the same year, proved with the Burali - Forti the class of all ordinals as non quantity. Although Cantor also described this antinomy, but not until 1899 in an unpublished letter. In it, he then presented the cardinal number - antinomy again as intensification of Burali - Forti paradox dar.

Second Cantor's antinomy

1899 showed Cantor via an indirect proof that " the epitome of everything thinkable " or " the system of all possible classes," the so-called universal class, no amount: If the universal class a lot, then the power set of the universal class would be a subset of the universal class and thus no powerful amount, as required by the theorem of Cantor. He proved that the universal class is a real class.