An Analysis of Tennenbaum’s Theorem in Constructive Type Theory

Authors Marc Hermes , Dominik Kirst

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Author Details

Marc Hermes
  • Department of Mathematics, Universität des Saarlandes, Saarbrücken, Germany
Dominik Kirst
  • Universität des Saarlandes, Saarland Informatics Campus, Saarbrücken, Germany

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Marc Hermes and Dominik Kirst. An Analysis of Tennenbaum’s Theorem in Constructive Type Theory. In 7th International Conference on Formal Structures for Computation and Deduction (FSCD 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 228, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Constructive mathematics
  • first-order logic
  • Peano arithmetic
  • Tennenbaum’s theorem
  • constructive type theory
  • Church’s thesis
  • synthetic computability
  • Coq


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