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：Tennenbaum定理指出，具有可计算算术运算的皮亚诺算术（PA）的唯一可数模型是自然数的标准模型。本文以构造类型理论为框架，重新审视、分析并推广了这一结果。所选框架支持可计算性理论的综合方法，利用构造类型理论中所有可定义函数在外部均能被证明为可计算这一特性。我们基于此视角展开研究，并通过假设丘奇论题的一个版本将其内化，该假设表达了自然数上的任意函数均可由PA中的公式表示。这一假设为进行严格的可计算性论证提供了便利的抽象设置，甚至在定理的形式化实现中亦如此。具体而言，我们对若干经典证明进行了构造化处理，并给出了Tennenbaum定理的一种固有构造性诠释，所有这些均源自文献中的论证。特别是在经典证明方面，构造性框架使我们能够突出其假设与结论中在经典视角下不可见的差异。所有版本均附有在Coq证明助手中的统一形式化实现。