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.

翻译：特南鲍姆定理指出，具有可计算算术运算的皮亚诺算术（PA）的唯可数模型是自然数的标准模型。本文以构造类型论为框架，重新审视、分析并推广了这一结果。所选框架允许对可计算性理论采用综合方法，利用外部视角下构造类型论中可定义的所有函数均可被证明为可计算这一性质。我们在此基础上进一步将这一观点内化，通过假设丘奇论题的一个版本（即自然数上的任何函数均可由PA中的公式表示）来实现。该假设为进行严格的可计算性论证（甚至包括定理的机械化实现）提供了便捷的抽象框架。具体而言，我们将若干经典证明构造化，并给出一种本质构造性的特南鲍姆定理阐释，所有内容均源自文献中的论证。针对经典证明，构造性设置使我们能够凸显其前提与结论中在经典框架下不可见的差异。所有版本均附带Coq证明助手中的统一机械化实现。