# An Analysis of Tennenbaum's Theorem in Constructive Type Theory

## TL;DR

This paper revisits Tennenbaum's theorem within constructive type theory, providing a synthetic computability framework, generalizations, and mechanized proofs in Coq, highlighting differences from classical assumptions.

## Contribution

It introduces a constructive approach to Tennenbaum's theorem, generalizes classical proofs, and provides a mechanized formalization in Coq.

## Key findings

- Constructive type theory enables a synthetic approach to computability.
- The paper generalizes classical Tennenbaum's theorem within a constructive setting.
- Mechanized proofs in Coq validate the constructive versions of the theorem.

## 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, analyze and generalize this result. The chosen framework allows for a synthetic approach to computability theory, exploiting that, externally, all functions definable in constructive type theory can be shown computable. We then build on this viewpoint, and furthermore internalize it by assuming a version of Church's thesis, which expresses that any function on natural numbers is representable by a formula in PA. This assumption provides for a conveniently abstract setup to carry out rigorous computability arguments, even in the theorem's 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.