Essential Incompleteness of Arithmetic Verified by Coq

  • Russell O’Connor
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3603)

Abstract

A constructive proof of the Gödel-Rosser incompleteness theorem [9] has been completed using Coq proof assistant. Some theory of classical first-order logic over an arbitrary language is formalized. A development of primitive recursive functions is given, and all primitive recursive functions are proved to be representable in a weak axiom system. Formulas and proofs are encoded as natural numbers, and functions operating on these codes are proved to be primitive recursive. The weak axiom system is proved to be essentially incomplete. In particular, Peano arithmetic is proved to be consistent in Coq’s type theory and therefore is incomplete.

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References

  1. 1.
    Burris, S.N.: Logic for mathematics and computer science: Supplementary text (1997), http://www.math.uwaterloo.ca/~snburris/htdocs/LOGIC/stext.html
  2. 2.
    Caprotti, O., Oostdijk, M.: Formal and efficient primality proofs by use of computer algebra oracles. J. Symb. Comput. 32(1/2), 55–70 (2001)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Despeyroux, J., Hirschowitz, A.: Higher-order abstract syntax with induction in coq. In: Pfenning, F. (ed.) LPAR 1994. LNCS, vol. 822, pp. 159–173. Springer, Heidelberg (1994)Google Scholar
  4. 4.
    Gödel, K.: Ueber Formal Unentscheidbare sätze der Principia Mathematica und Verwandter Systeme I. Monatshefte für Mathematik und Physik 38, 173–198 (1931); english translation: On Formally Undecidable Propositions of Principia Mathematica and Related Systems I. Oliver & Boyd, London (1962)CrossRefGoogle Scholar
  5. 5.
    Harrison, J.: Formalizing basic first order model theory. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998, vol. 1479, pp. 153–170. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Harrison, J.: The HOL-Light manual (2000)Google Scholar
  7. 7.
    Marche, C.: Fwd: Question about fixpoint. Coq club mailing list correspondence [cited 2005-02-07] (February 2005), http://pauillac.inria.fr/pipermail/coq-club/2005/001641.html
  8. 8.
    McBride, C.: Dependently Typed Functional Programs and their Proofs. PhD thesis, University of Edinburgh (1999), Available from http://www.lfcs.informatics.ed.ac.uk/reports/00/ECS-LFCS-00-419/
  9. 9.
    O’Connor, R.: The Gödel-Rosser 1st incompleteness theorem (March 2005), http://r6.ca/Goedel20050512.tar.gz
  10. 10.
    Hodel, R.E.: An Introduction to Mathematical Logic. PWS Pub. Co. (1995)Google Scholar
  11. 11.
    Shankar, N.: Metamathematics, Machines, and Gödel’s Proof. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (1994)Google Scholar
  12. 12.
    Shoenfield, J.R.: Mathematical Logic. Addison-Wesley, Reading (1967)MATHGoogle Scholar
  13. 13.
    Stoughton, A.: Substitution revisited 59(3), 317–325 (August 1988)Google Scholar
  14. 14.
    The Coq Development Team. The Coq Proof Assistant Reference Manual – Version V8.0 (April 2004), http://coq.inria.fr

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Russell O’Connor
    • 1
    • 2
  1. 1.Institute for Computing and Information Science, Faculty of ScienceRadboud UniversityNijmegen
  2. 2.The Group in Logic and the Methodology of ScienceUniversity of CaliforniaBerkeley

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