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。所选框架支持对可计算性理论采取综合方法，利用了外部而言构造类型理论中所有可定义函数均可证明是可计算的这一特性。在此基础上，我们进一步内化这一观点，假设丘奇论题的一种版本成立，该版本断言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（包括定理的机械化实现）提供了便捷的抽象框架。具体而言，我们构造化了多个经典证明，并给出了Tennenbaum定理的一种内在构造性阐述，所有工作均遵循文献中的论证思路。特别地，在经典证明方面，构造性设定使我们能够凸显出经典视角下不可见的假设与结论之间的差异。所有版本均在Coq证明助手中实现了统一的机械化。