An Analysis of Tennenbaum’s Theorem in Constructive Type Theory

Marc Hermes, Dominik Kirst

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

3 Scopus citations

Abstract

Tennenbaum’s theorem states that the only countable model of Peano arithmetic (PA) with computable arithmetical operations is the standard model of natural numbers. In this paper, we use constructive type theory as a framework to revisit and generalize this result. The chosen framework allows for a synthetic approach to computability theory, by exploiting the fact that, externally, all functions definable in constructive type theory can be shown computable. We internalize this fact by assuming a version of Church’s thesis expressing that any function on natural numbers is representable by a formula in PA. This assumption allows for a conveniently abstract setup to carry out rigorous computability arguments and feasible mechanization. Concretely, we constructivize several classical proofs and present one inherently constructive rendering of Tennenbaum’s theorem, all following arguments from the literature. Concerning the classical proofs in particular, the constructive setting allows us to highlight differences in their assumptions and conclusions which are not visible classically. All versions are accompanied by a unified mechanization in the Coq proof assistant.

Original languageEnglish
Title of host publication7th International Conference on Formal Structures for Computation and Deduction, FSCD 2022
EditorsAmy P. Felty
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772334
DOIs
StatePublished - 1 Jun 2022
Externally publishedYes
Event7th International Conference on Formal Structures for Computation and Deduction, FSCD 2022 - Haifa, Israel
Duration: 2 Aug 20225 Aug 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume228
ISSN (Print)1868-8969

Conference

Conference7th International Conference on Formal Structures for Computation and Deduction, FSCD 2022
Country/TerritoryIsrael
CityHaifa
Period2/08/225/08/22

Keywords

  • Church’s thesis
  • Coq
  • Peano arithmetic
  • Tennenbaum’s theorem
  • constructive type theory
  • first-order logic
  • synthetic computability

ASJC Scopus subject areas

  • Software

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