A Finite First-Order Theory of Classes

  • Florent Kirchner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4502)


We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. These axioms have the property of being easily orientable into rewrite rules. This allows us to give finite first-order axiomatizations of arithmetic and real fields theory, and a presentation of arithmetic in deduction modulo that has a finite number of rewrite rules. Overall, this formalization relies on a weak calculus of explicit substitutions to provide a simple and finite framework.


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  1. 1.
    von Neumann, J.: Eine Axiomatisierung der Mengenlehre. Journal für die reine und angewandte Mathematik 154, 219–240 (1925)CrossRefGoogle Scholar
  2. 2.
    Gödel, K.: The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory. In: Annals of Mathematics Studies, vol. 3, Princeton University Press, Princeton (1940)Google Scholar
  3. 3.
    Bernays, P.: Axiomatic Set Theory. Dover Publications (1958)Google Scholar
  4. 4.
    Mendelson, E.: Introduction to mathematical logic, 4th edn. Chapman & Hall, Sydney (1997)MATHGoogle Scholar
  5. 5.
    Vaillant, S.: A finite first-order presentation of set theory. In: Geuvers, H., Wiedijk, F. (eds.) TYPES 2002. LNCS, vol. 2646, pp. 316–330. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Kirchner, F.: Fellowship: who needs a manual anyway? (2005)Google Scholar
  7. 7.
    Ridge, T., Margetson, J.: A mechanically verified, sound and complete theorem prover for first order logic. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 294–309. Springer, Heidelberg (2005)Google Scholar
  8. 8.
    Dowek, G., Hardin, T., Kirchner, C.: HOL-λσ: An intentional first-order expression of higher-order logic. Mathematical Structures in Computer Science 11(1), 21–45 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hardin, T., Maranget, L., Pagano, B.: Functional back-ends within the lambda-sigma calculus. In: ICFP, pp. 25–33. ACM Press, New York (1996)Google Scholar
  10. 10.
    Dowek, G., Miquel, A.: Cut elimination for Zermelo’s set theory. Submitted to RTA 2006 (2006)Google Scholar
  11. 11.
    Dowek, G., Werner, B.: Arithmetic as a theory modulo. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 423–437. Springer, Heidelberg (2005)Google Scholar
  12. 12.
    Dowek, G., Hardin, T., Kirchner, C.: Theorem proving modulo. Journal of Automated Reasoning 31(1), 33–72 (2003)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Lelong-Ferrand, J., Arnaudies, J.M.: Cours de Mathématiques. Tome 2 : Analyse. Dunod (1972)Google Scholar
  14. 14.
    Bourbaki, N.: Éléments de mathématique – Théorie des ensembles. vol. 1 à 4. Masson, Paris (1968)Google Scholar
  15. 15.
    Megill, N.: A finitely axiomatized formalization of predicate calculus with equality. Notre Dame Journal of Formal Logic 36(3), 435–453 (1995)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Belinfante, J.: Computer proofs in Gödel’s class theory with equational definitions for composite and cross. Journal of Automated Reasoning 22(2), 311–339 (1999)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Boyer, R., Lusk, E., McCune, W., Overbeek, R., Stickel, M., Wos, L.: Set theory in first-order logic: Clauses for Gödel’s axioms. Journal of Automated Reasoning 2(3), 287–327 (1986)MATHCrossRefGoogle Scholar
  18. 18.
    Quaife, A.: Automated deduction in von Neumann-Bernays-Gödel set theory. Journal of Automated Reasoning 8(1), 91–147 (1992)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Florent Kirchner
    • 1
  1. 1.LIX, École Polytechnique, 91128 PalaiseauFrance

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