Tham khảo Hệ tiên đề Peano

Trích dẫn

  1. ↑ Grassmann 1861.
  2. ↑ Peirce 1881, Shields 1997
  3. ↑ van Heijenoort 1967, tr. 94.
  4. ↑ van Heijenoort 1967, tr. 2.
  5. 1 2 van Heijenoort 1967, tr. 83.
  6. ↑ Peano 1889, tr. 1.
  7. ↑ Partee, Ter Meulen & Wall 2012, tr. 215.
  8. ↑ Harsanyi (1983).
  9. ↑ Mendelson 1997, tr. 155.
  10. ↑ Kaye 1991, tr. 16–18.
  11. ↑ Suppes 1960, Hatcher 1982
  12. ↑ Tarski & Givant 1987, Section 7.6.
  13. ↑ Hermes 1973, VI.4.3, presenting a theorem of Thoralf Skolem
  14. ↑ Hermes 1973, VI.3.1.
  15. ↑ Kaye 1991, Section 11.3.
  16. ↑ Kaye 1991, tr. 70ff..
  17. ↑ Fritz 1952, p. 137An illustration of 'interpretation' is Russell's own definition of 'cardinal number'. The uninterpreted system in this case is Peano's axioms for the number system, whose three primitive ideas and five axioms, Peano believed, were sufficient to enable one to derive all the properties of the system of natural numbers. Actually, Russell maintains, Peano's axioms define any progression of the form x 0 , x 1 , x 2 , … , x n , … {\displaystyle x_{0},x_{1},x_{2},\ldots ,x_{n},\ldots } of which the series of the natural numbers is one instance.
  18. ↑ Gray 2013, p. 133So Poincaré turned to see whether logicism could generate arithmetic, more precisely, the arithmetic of ordinals. Couturat, said Poincaré, had accepted the Peano axioms as a definition of a number. But this will not do. The axioms cannot be shown to be free of contradiction by finding examples of them, and any attempt to show that they were contradiction-free by examining the totality of their implications would require the very principle of mathematical induction Couturat believed they implied. For (in a further passage dropped from S&M) either one assumed the principle in order to prove it, which would only prove that if it is true it is not self-contradictory, which says nothing; or one used the principle in another form than the one stated, in which case one must show that the number of steps in one's reasoning was an integer according to the new definition, but this could not be done (1905c, 834).
  19. ↑ Hilbert 1902.
  20. ↑ Gödel 1931.
  21. ↑ Gödel 1958
  22. ↑ Gentzen 1936
  23. ↑ Willard 2001.

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WikiPedia: Hệ tiên đề Peano http://www.math.uwaterloo.ca/~snburris/htdocs/scav... http://mathworld.wolfram.com/.html http://digisrv-1.biblio.etc.tu-bs.de:8080/docporta... http://www.uni-potsdam.de/u/philosophie/grassmann/... http://www.w-k-essler.de/pdfs/goedel.pdf http://www.utm.edu/research/iep/p/poincare.htm http://www.ams.org/journals/bull/1902-08-10/S0002-... //www.ams.org/mathscinet-getitem?mr=1507856 //www.ams.org/mathscinet-getitem?mr=1833464 //dx.doi.org/10.1007%2F978-94-015-7676-5_8