Church's thesis and related axioms in Coq's type theory

TL;DR

This paper investigates the implications of adopting Church's thesis as an axiom within Coq's constructive type theory, analyzing its consistency and impact on classical and choice-related axioms, supported by mechanized proofs.

Contribution

It provides a detailed analysis of Church's thesis in Coq's type theory, exploring which axioms preserve computational intuition and which lead to inconsistency, with all results mechanized in Coq.

Findings

Church's thesis is consistent with constructive type theory.

Certain classical and choice axioms conflict with Church's thesis.

Mechanized proofs in Coq validate the analysis.

Abstract

"Church's thesis" ($\mathsf{CT}$) as an axiom in constructive logic states that every total function of type $\mathbb{N} \to \mathbb{N}$ is computable, i.e. definable in a model of computation. $\mathsf{CT}$ is inconsistent in both classical mathematics and in Brouwer's intuitionism since it contradicts Weak K\"onig's Lemma and the fan theorem, respectively. Recently, $\mathsf{CT}$ was proved consistent for (univalent) constructive type theory. Since neither Weak K\"onig's Lemma nor the fan theorem are a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that $\mathsf{CT}$ is inconsistent only with a combination of classical logic and choice axioms. We study consequences of $\mathsf{CT}$ and its relation to several classes of axioms in Coq's type theory, a constructive type theory with a universe of propositions which does neither prove classical logical axioms nor strong choice axioms. We thereby provide a partial answer to the question which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant.