Burali-Forti paradox
The Burali-Forti paradox is the oldest paradox of naive set theory , published on March 28, 1897. It describes the contradiction in which the formation of the set of all ordinal numbers fails. It is named after its discoverer Cesare Burali-Forti , who showed that such a set of all ordinal numbers would itself correspond to an ordinal number to which a larger successor ordinal number could be formed, which would be less than or equal , from which the impossible inequality followed.
Georg Cantor only described the paradox in 1899 as a generalization of the first Cantorian antinomy , with which he proved that the class of all cardinal numbers is not a set. This class can be seen as a real sub-class of ordinal numbers.
In the axiomatic Zermelo set theory or Zermelo-Fraenkel set theory (ZF), the Burali-Forti paradox can be understood as proof that no set of all ordinal numbers exists. In set theories that work with classes , it proves that the class of all ordinals is a real class.
See also
literature
- C. Burali-Forti: Una questione sui numeri transfiniti. In: Rendiconti del Circolo Matematico di Palermo. Vol. 11, 1897, ISSN 0009-725X , pp. 154-164, digitized . English translation: A question on transfinite numbers. In: Jean van Heijenoort : From Frege to Gödel. A Source Book in Mathematical Logic, 1879-1931. Harvard University Press, Cambridge MA et al. 1967, pp. 104-112.
- Infinite (plus one). Hilbert Hotel, Russell's Barber, Peano's Ladder to Heaven, Cantor's Diagonal, Planck's Constant. (= Spectrum of Science. Special 2, 2005). Spektrum der Wissenschaft Verlagsgesellschaft, Heidelberg, ISBN 3-938639-08-3 , p. 36.
Web links
Individual evidence
- ^ Letter of August 3, 1899 to Richard Dedekind in: Georg Cantor: Briefe. Edited by Herbert Meschkowski and Winfried Nilson. Springer, Berlin et al. 1991, ISBN 3-540-50621-7 , p. 408. Earlier dates are often mentioned, but no sources exist for them.