Escaping Tennenbaum's Theorem and a Strong Jump Inversion Theorem

TL;DR

This paper demonstrates the fragility of Tennenbaum's theorem for intermediate fragments of PA by constructing theories with computable nonstandard models and introduces a general strong jump inversion theorem.

Contribution

It constructs theories equivalent to PA plus all $ ext{Pi}^0_n$ truths that admit computable nonstandard models and develops a broad strong jump inversion theorem applicable to multiple results.

Findings

Constructed theories equivalent to PA plus all $ ext{Pi}^0_n$ truths with computable nonstandard models

Developed a general strong jump inversion theorem

Reinterpreted known results as applications of the new theorem

Abstract

Tennenbaum's theorem states that PA does not admit any nonstandard computable model. In 2022, Pakhomov proved that this theorem is fragile in regards to how PA is expressed, by constructing a theory that is definitionally equivalent to PA (roughly: "it's PA but with a different choice of signature") for which there is a computable nonstandard model. He showed that this fragility does not extend to true arithmetic (any nonstandard model of a theory definitionally equivalent to $\mathrm{Th}(\mathbb{N})$ is not computable), but the question of whether this fragility extends to fragments of PA of intermediate strength was left open. We show that it does, by constructing a sequence of theories $T^n$ which are definitionally equivalent to: "PA plus all $\Pi^0_n$ truths", all of which admit computable nonstandard models. In the process, we produce a general-purpose theorem for strong jump inversion. Besides applying this theorem to obtain our novel result, we show that several known results from the literature can be seen as direct applications of our theorem.