Essential Incompleteness of Arithmetic Verified by Coq

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


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.


Function Symbol Axiom System Recursive Call Chinese Remainder Theorem Internal Logic 
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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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