Cantor's antinomy

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Georg Cantor described several antinomies between 1897 and 1899 , by means of which he proved that certain classes are not sets. His evidence shows that he did not have a naively contradictory concept of sets, which is often claimed because of Cantor's definition of sets . He already separated sets as consistent multiplicities from inconsistent multiplicities, which are now called real classes . Because Cantor did not publish his antinomies, but only communicated them by letter to David Hilbert and Richard Dedekind , his set theory was often wrongly assessed as naive set theory . It was not until Zermelo published his letters in 1932 that it became known that the inventor of set theory was aware of the antinomy problem very early on. Cantor's axioms of sets from these very letters constitute the oldest notable solution to the problem.

First Cantor antinomy

In 1897 Cantor showed that the class of all (transfinite) cardinal numbers , the “totality of all alephs ”, is not a set, but a real class, via an indirect proof : If this totality were a set, there would be a larger aleph, the element would and would not belong to this totality. Cantor only became aware of this antinomy in the last few years: in 1890 he declared the “epitome of all power” as a “well-ordered set”.

The first Cantor antinomy is to be distinguished from the Burali-Forti paradox from the same year with which Burali-Forti demonstrated the class of all ordinal numbers as a non-set. Cantor also described this antinomy, but not until 1899 in an unpublished letter. In it he then presented the cardinal number antinomy again as an intensification of the Burali-Forti paradox.

Second Cantor Antinomy

In 1899 Cantor showed via an indirect proof that “the epitome of everything thinkable” or “the system of all thinkable classes”, the so-called universal class , is not a set: If the universal class were a set, then the power set of the universal class would be a subset of the universal class and thus not a more powerful set, as Cantor's theorem requires. With this he proved that the all-class is a real class.

Individual evidence

  1. ^ Letter from Cantor to Dedekind of August 3, 1899. In: Georg Cantor: Briefe. Edited by Herbert Meschkowski and Winfried Nilson. Springer, Berlin et al. 1991, ISBN 3-540-50621-7 , p. 407.
  2. ^ Letter from Cantor to Hilbert of September 26, 1897. In: Georg Cantor: Briefe. Edited by Herbert Meschkowski and Winfried Nilson. Springer, Berlin et al. 1991, ISBN 3-540-50621-7 , p. 388.
  3. ↑ Georg Cantor: On an elementary question of the theory of manifolds . In: German Mathematicians Association (ed.): Annual report of the German Mathematicians Association . tape 1 . Reimer, 1892, ISSN  0012-0456 , p. 75–78 ( uni-goettingen.de - quote on p. 77 below).
  4. ^ Letter from Cantor to Dedekind of August 3, 1899. In: Georg Cantor: Briefe. Edited by Herbert Meschkowski and Winfried Nilson. Springer, Berlin et al. 1991, ISBN 3-540-50621-7 , p. 408.
  5. ^ Letter from Cantor to Dedekind of August 3, 1899 and August 30, 1899. In: Georg Cantor: Collected treatises of mathematical and philosophical content. With explanatory notes and additions from the Cantor-Dedekind correspondence. Published by Ernst Zermelo . Springer, Berlin 1932, p. 448 (system of all conceivable classes), and in: Georg Cantor: Briefe. Edited by Herbert Meschkowski and Winfried Nilson. Springer, Berlin et al. 1991, ISBN 3-540-50621-7 , p. 407 (epitome of everything imaginable).