Essential Incompleteness of Arithmetic Verified by Coq

  • Russell O’Connor
Conference paper

DOI: 10.1007/11541868_16

Part of the Lecture Notes in Computer Science book series (LNCS, volume 3603)
Cite this paper as:
O’Connor R. (2005) Essential Incompleteness of Arithmetic Verified by Coq. In: Hurd J., Melham T. (eds) Theorem Proving in Higher Order Logics. TPHOLs 2005. Lecture Notes in Computer Science, vol 3603. Springer, Berlin, Heidelberg


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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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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