Theorem Proving in Higher Order Logics

Volume 3603 of the series Lecture Notes in Computer Science pp 245-260

Essential Incompleteness of Arithmetic Verified by Coq

  • Russell O’ConnorAffiliated withInstitute for Computing and Information Science, Faculty of Science, Radboud UniversityThe Group in Logic and the Methodology of Science, University of California

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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.