Synthetic undecidability and incompleteness of first-order axiom systems in Coq

Dominik Kirst, Marc Hermes

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations

Abstract

We mechanise the undecidability of various first-order axiom systems in Coq, employing the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments of Peano arithmetic (PA) and Zermelo-Fraenkel set theory (ZF), with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e. Hilbert's tenth problem (H10), and the Post correspondence problem (PCP), respectively. In the synthetic setting based on the computability of all functions definable in a constructive foundation, such as Coq's type theory, it suffices to define these reductions as meta-level functions with no need for further encoding in a formalised model of computation. The concrete cases of PA and ZF are prepared by a general synthetic theory of undecidable axiomatisations, focusing on well-known connections to consistency and incompleteness. Specifically, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic extensions still justified by such standard models are shown incomplete. As a by-product of the undecidability of ZF formulated using only membership and no equality symbol, we obtain the undecidability of first-order logic with a single binary relation.

Original languageEnglish
Title of host publication12th International Conference on Interactive Theorem Proving, ITP 2021
EditorsLiron Cohen, Cezary Kaliszyk
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771887
DOIs
StatePublished - 1 Jun 2021
Externally publishedYes
Event12th International Conference on Interactive Theorem Proving, ITP 2021 - Virtual, Rome, Italy
Duration: 29 Jun 20211 Jul 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume193
ISSN (Print)1868-8969

Conference

Conference12th International Conference on Interactive Theorem Proving, ITP 2021
Country/TerritoryItaly
CityVirtual, Rome
Period29/06/211/07/21

Keywords

  • Constructive type theory
  • Coq
  • First-order logic
  • Incompleteness
  • Peano arithmetic
  • Synthetic computability
  • Undecidability
  • ZF set theory

ASJC Scopus subject areas

  • Software

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