Journal of Automated Reasoning

, Volume 23, Issue 3, pp 197–234 | Cite as

On Equivalents of Well-Foundedness

  • Piotr Rudnicki
  • Andrzej Trybulec
Article

Abstract

Four statements equivalent to well-foundedness (well-founded induction, existence of recursively defined functions, uniqueness of recursively defined functions, and absence of descending ω-chains) have been proved in Mizar and the proofs mechanically checked for correctness. It seems not to be widely known that the existence (without the uniqueness assumption) of recursively defined functions implies well-foundedness. In the proof we used regular cardinals, a fairly advanced notion of set theory. The theory of cardinals in Mizar was developed earlier by G. Bancerek. With the current state of the Mizar system, the proofs turned out to be an exercise with only minor additions at the fundamental level. We would like to stress the importance of a systematic development of a mechanized data base for mathematics in the spirit of the QED Project. 12pt ENOD – Experience, Not Only Doctrine

G. Kreisel

Mizar QED Project set theory well-foundedness regular cardinals 

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References

  1. 1.
    Bancerek, G.: Directed sets, nets, ideals, filters, and maps, Formalized Mathematics, 1997, to appear.Google Scholar
  2. 2.
    Bancerek, G.: Cardinal numbers, Formalized Mathematics 1(2) (1990), 377-382.Google Scholar
  3. 3.
    Bancerek, G.: The ordinal numbers, Formalized Mathematics 1(1) (1990), 91-96.Google Scholar
  4. 4.
    Bancerek, G.: Sequences of ordinal numbers, Formalized Mathematics 1(2) (1990), 281-290.Google Scholar
  5. 5.
    Bancerek, G.: The well ordering relations, Formalized Mathematics 1(1) (1990), 123-129.Google Scholar
  6. 6.
    Bancerek, G.: Zermelo theorem and axiom of choice, Formalized Mathematics 1(2) (1990), 265-267.Google Scholar
  7. 7.
    Bancerek, G.: Countable sets and Hessenberg' theorem, Formalized Mathematics 2(1) (1991), 65-69.Google Scholar
  8. 8.
    Bancerek, G.: nig' lemma, Formalized Mathematics 2(3) (1991), 397-402.Google Scholar
  9. 9.
    Bancerek, G.: On powers of cardinals, Formalized Mathematics 3(1) (1992), 89-93.Google Scholar
  10. 10.
    Bancerek, G. and Hryniewiecki, K.: Segments of natural numbers and finite sequences, Formalized Mathematics 1(1) (1990), 107-114.Google Scholar
  11. 11.
    Bancerek, G. and Trybulec, A.: Miscellaneous facts about functions, Formalized Mathematics 5(4) (1996), 485-492.Google Scholar
  12. 12.
    Białas, J.: Group and field definitions, Formalized Mathematics 1(3) (1990), 433-439.Google Scholar
  13. 13.
    Białas, J. and Nakamura, Y.: The theorem ofWeierstrass, Formalized Mathematics 5(3) (1996), 353-359.Google Scholar
  14. 14.
    Boyer, R. S.: A mechanically proof-checked Encyclopedia of Mathematics: Should we build one? Can we? in A. Bundy (ed.), 12th International Conference on Automated Deduction LNAI/LNCS 814, Springer-Verlag, 1994, pp. 237-251.Google Scholar
  15. 15.
    Byliński, C.: A classical first order language, Formalized Mathematics 1(4) (1990), 669-676.Google Scholar
  16. 16.
    Bylińnski, C.: Functions and their basic properties, Formalized Mathematics 1(1) (1990), 55-65.Google Scholar
  17. 17.
    Bylińnski, C.: Partial functions, Formalized Mathematics 1(2) (1990), 357-367.Google Scholar
  18. 18.
    Bylińnski, C.: Some basic properties of sets, Formalized Mathematics 1(1) (1990), 47-53.Google Scholar
  19. 19.
    Darmochwał, A.: Finite sets, Formalized Mathematics 1(1) (1990), 165-167.Google Scholar
  20. 20.
    Darmochwał, A.: Calculus of quantifiers: Deduction theorem, Formalized Mathematics 2(2) (1991), 309-312.Google Scholar
  21. 21.
    Darmochwał, A. and Nakamura, Y.: Heine-Borel' covering theorem, FormalizedMathematics 2(4) (1991), 609-610.Google Scholar
  22. 22.
    de la Cruz, A.: Fix point theorem for compact spaces, Formalized Mathematics 2(4) (1991), 505-506.Google Scholar
  23. 23.
    Franzén, T.: Teaching mathematics through formalism: A few caveats, in D. Gries (ed.), Proceedings of the DIMACS Symposium on Teaching Logic, DIMACS, 1996. On WWW: http://dimacs.rutgers.edu/Workshops/Logic/program.html.Google Scholar
  24. 24.
    Gries, D. and Schneider, F. B.: A Logical Approach to Discrete Math., Springer-Verlag, 1994.Google Scholar
  25. 25.
    Harrison, J.: Inductive definitions: Automation and application, in Proceedings of the 1995 International Workshop on Higher Order Logic Theorem Proving and Its Applications, Lecture Notes in Comput. Sci. 971, Springer-Verlag, 1995, pp. 200-213.Google Scholar
  26. 26.
    Harrison, J.: A Mizar mode for HOL, in J. von Wright, J. Grundy, and J. Harrison (eds.), Theorem Proving in Higher Order Logics: 9th International Conference, TPHOLs'96, Turku, Finland, Lecture Notes in Comput. Sci. 1125, Springer-Verlag, 1996, pp. 203-220.Google Scholar
  27. 27.
    Hayden, S. and Kennison, J. F.: Zermelo Fraenkel Set Theory, Charles E. Merrill Publishing Co., Columbus, Ohio, 1968.Google Scholar
  28. 28.
    Hryniewiecki, K.: Basic properties of real numbers, Formalized Mathematics 1(1) (1990), 35-40.Google Scholar
  29. 29.
    Hryniewiecki, K.: Recursive definitions, Formalized Mathematics 1(2) (1990), 321-328.Google Scholar
  30. 30.
    Jaśkowski, S.: On the rules of supposition in formal logic, Studia Logica 1 (1934).Google Scholar
  31. 31.
    Korolkiewicz, M.: The de l'Hospital theorem, Formalized Mathematics 2(5) (1991), 675-678.Google Scholar
  32. 32.
    Kotowicz, J., Raczkowski, K. and Sadowski, P.: Average value theorems for real functions of one variable, Formalized Mathematics 1(4) (1990), 803-805.Google Scholar
  33. 33.
    Kusak, E.: Desargues theorem in projective 3-space, Formalized Mathematics 2(1) (1991), 13-16.Google Scholar
  34. 34.
    Lamport, L.: How to write a proof, Research Report 94, DEC Systems Reserach Center, Palo Alto, CA, 1993.Google Scholar
  35. 35.
    McCall, S. (ed.): Polish Logic in 1920-1939, Clarendon Press, Oxford, 1967.Google Scholar
  36. 36.
    Nowak, B. and Trybulec, A.: Hahn-Banach theorem, Formalized Mathematics 4(1) (1993), 29-34.Google Scholar
  37. 37.
    Ono, K.: On a practical way of describing formal deductions, Nagoya Math. J. 21 (1962).Google Scholar
  38. 38.
    Popiołek, J.: Real normed space, Formalized Mathematics 2(1) (1991), 111-115.Google Scholar
  39. 39.
    Rudnicki, P. and Trybulec, A.: How to write a proof, Technical Report 96-08, University of Alberta, Department of Computing Science, 1996. 19 pages.Google Scholar
  40. 40.
    Rudnicki, P. and Trybulec, A.: Fix-points in complete lattices, Formalized Mathematics (1997), to appear.Google Scholar
  41. 41.
    Tarski, A.: On well-ordered subsets of any set, Fund. Math. 32 (1939), 176-183.Google Scholar
  42. 42.
    Trybulec, A.: Built-in concepts, Formalized Mathematics 1(1) (1990), 13-15.Google Scholar
  43. 43.
    Trybulec, A.: Function domains and Frænkel operator, Formalized Mathematics 1(3) (1990), 495-500.Google Scholar
  44. 44.
    Trybulec, A.: Tarski Grothendieck set theory, Formalized Mathematics 1(1) (1990), 9-11.Google Scholar
  45. 45.
    Trybulec, W. A.: Partially ordered sets, Formalized Mathematics 1(2) (1990), 313-319.Google Scholar
  46. 46.
    Trybulec, W. A.: Subgroup and cosets of subgroups, Formalized Mathematics 1(5) (1990), 855-864.Google Scholar
  47. 47.
    Trybulec, W. A. and Bancerek, G.: Kuratowski-Zorn lemma, Formalized Mathematics 1(2) (1990), 387-393.Google Scholar
  48. 48.
    Trybulec, Z. and Święczkowska, H.: Boolean properties of sets, Formalized Mathematics 1(1) (1990), 17-23.Google Scholar
  49. 49.
    Walijewski, J. S.: Representation theorem for Boolean algebras, Formalized Mathematics 4(1) (1993), 45-50.Google Scholar
  50. 50.
    Watanabe, T.: The Brouwer fixed point theorem for intervals, Formalized Mathematics 3(1) (1992), 85-88.Google Scholar
  51. 51.
    Woronowicz, E.: Relations and their basic properties, Formalized Mathematics 1(1) (1990), 73-83.Google Scholar
  52. 52.
    Woronowicz, E.: Relations defined on sets, Formalized Mathematics 1(1) (1990), 181-186.Google Scholar
  53. 53.
    Żynel, M.: The Steinitz theorem and the dimension of a vector space, Formalized Mathematics 5(3) (1996), 423-428.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Piotr Rudnicki
    • 1
  • Andrzej Trybulec
    • 2
  1. 1.Department of Computing ScienceUniversity of AlbertaEdmontonCanada
  2. 2.Institute of Mathematics, Warsaw University in BiałystokPoland

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