Advertisement

Journal of Automated Reasoning

, Volume 17, Issue 3, pp 291–323 | Cite as

Mechanizing set theory

Cardinal arithmetic and the Axiom of Choice
  • Lawrence C. Paulson
  • Krzysztof Grabczewski
Article

Abstract

Fairly deep results of Zermelo-Frænkel (ZF) set theory have been mechanized using the proof assistant Isabelle. The results concern cardinal arithmetic and the Axiom of Choice (AC). A key result about cardinal multiplications is κ ⊗ κ=κ, where κ is any infinite cardinal. Proving this result required developing theories of orders, order-isomorphisms, order types, ordinal arithmetic, cardinals, etc.; this covers most of Kunen, Set Theory, Chapter I. Furthermore, we have proved the equivalence of 7 formulations of the Well-ordering Theorem and 20 formulations of AC; this covers the first two chapters of Rubin and Rubin, Equivalents of the Axiom of Choice, and involves highly technical material. The definitions used in the proofs are largely faithful in style to the original mathematics.

Key words

Isabelle cardinal arithmetic Axiom of Choice set theory QED project 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abrial, J. R. and Laffitte, G.: Towards the mechanization of the proofs of some classical theorems of set theory, Preprint, 1993.Google Scholar
  2. 2.
    Bancerek, Grzegorz: Countable sets and Hessenberg's theorem, Formalized Mathematics 2 (1990), 499–503. On the World Wide Web at http://math.uw.bialystok.pl/ ∼Form.-Math/Vol2/dvi/card_4.dvi.Google Scholar
  3. 3.
    Boyer, R. S. and Strother Moore, J.: A Computational Logic, Academic Press, 1979.Google Scholar
  4. 4.
    Boyer, R. S. and Strother Moore, J.: A Computational Logic Handbook, Academic Press, 1988.Google Scholar
  5. 5.
    Gilles Dowek: The Coq Proof Assistant User's Guide, Technical Report 154, INRIA-Rocquencourt, 1993.Google Scholar
  6. 6.
    Farmer, W. M., Guttman, J. D., and Thayer, J. F.: IMPS: An interactive mathematical proof system, J. Automated Reasoning 11(2) (1993), 213–248.Google Scholar
  7. 7.
    Gardner, M.: The Unexpected Hanging and Other Mathematical Diversions, University of Chicago Press, 1991.Google Scholar
  8. 8.
    Gordon, M. J. C. and Melham, T. F.: Introduction to HOL: A Theorem Proving Environment for Higher Order Logic, Cambridge University Press, 1993.Google Scholar
  9. 9.
    Halmos, P. R.: Naive Set Theory, Van Nostrand, 1960.Google Scholar
  10. 10.
    Huet, G.: Residual theory in λ-calculus: A formal development, Journal of Functional Programming 4(3) (1994), 371–394.Google Scholar
  11. 11.
    van Bentham Jutting, L. S.: Checking Landau's ‘Grundlagen’ in the AUTOMATH System, PhD thesis, Eindhoven University of Technology, 1977.Google Scholar
  12. 12.
    Kunen, K.: Set Theory: An Introduction to Independence Proofs, North-Holland, 1980.Google Scholar
  13. 13.
    Miller, D.: Unification under a mixed prefix, Journal of Symbolic Computation 14(4) (1992), 321–358.Google Scholar
  14. 14.
    Nederpelt, R. P., Geuvers, J. H., and de Vrijer, R. C.: Selected Papers on Automath, North-Holland, 1994.Google Scholar
  15. 15.
    Noël, Ph.: Experimenting with Isabelle in ZF set theory, J. Automated Reasoning 10(1) (1993), 15–58.Google Scholar
  16. 16.
    Paulson, L. C.: Constructing recursion operators in intuitionistic type theory, Journal of Symbolic Computation 2 (1986), 325–355.Google Scholar
  17. 17.
    Paulson, L. C.: Isabelle: The next 700 theorem provers, in P. Odifreddi (ed.), Logic and Computer Science, Academic Press, 1990, pp. 361–386.Google Scholar
  18. 18.
    Paulson, L. C.: Set theory for verification: I. From foundations to functions, J. Automated Reasoning 11(3) (1993), 353–389.Google Scholar
  19. 19.
    Paulson, L. C.: A fixedpoint approach to implementing (co)inductive definitions, in Alan Bundy (ed.), Automated Deduction — CADE-12, LNAI 814, Springer, 1994, pp. 148–161, 12th international conference.Google Scholar
  20. 20.
    Paulson, L. C.: Isabelle: A Generic Theorem Prover, LNCS 828, Springer, 1994.Google Scholar
  21. 21.
    Paulson, L. C.: Set theory for verification: II. Induction and recursion, J. Automated Reasoning 15(2) (1995), 167–215.Google Scholar
  22. 22.
    The QED manifesto: On the World Wide Web at http://www.mcs.anl.gov/home/lusk/qed/manifesto.html, 1995.Google Scholar
  23. 23.
    Art Quaife: Automated deduction in von Neumann-Bernays-Gödel set theory, J. Automated Reasoning 8(1) (1992), 91–147.Google Scholar
  24. 24.
    Rubin, H. and Rubin, J. E.: Equivalents of the Axiom of Choice, II, North-Holland, 1985.Google Scholar
  25. 25.
    Russinoff, D. M.: A mechanical proof of quadratic reciprocity, J. Automated Reasoning 8(1) (1992), 3–22.Google Scholar
  26. 26.
    Shankar, N.: Metamathematics, Machines, and Gödel's Proof, Cambridge University Press, 1994.Google Scholar
  27. 27.
    Suppes, P.: Axiomatic Set Theory, Dover, 1972.Google Scholar
  28. 28.
    Yu, Yuan: Computer proofs in group theory, J. Automated Reasoning 6(3) (1990), 251–286.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Lawrence C. Paulson
    • 1
  • Krzysztof Grabczewski
    • 2
  1. 1.Computer LaboratoryUniversity of CambridgeUK
  2. 2.Nicholas Copernicus UniversityToruńPoland

Personalised recommendations