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Escaping Tennenbaum's Theorem and a Strong Jump Inversion Theorem

Duarte Maia

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 $Π^0_n$ 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.

Escaping Tennenbaum's Theorem and a Strong Jump Inversion 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 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 which are definitionally equivalent to: "PA plus all 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.
Paper Structure (18 sections, 24 theorems, 14 equations)

This paper contains 18 sections, 24 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Background -- Tennenbaum's Theorem
  3. Pakhomov's Construction and New Results
  4. A Strong Jump Inversion Theorem
  5. A General Jump Inversion Theorem
  6. C.e.-typed Structures
  7. The QETP
  8. Main Result
  9. Previously Known Applications
  10. Equivalence Relations of Infinite Rank
  11. Equivalence Relations of Finite Rank
  12. Khisamiev's Theorem
  13. Linear Orders with Successors and Predecessors
  14. A Simple Class of Trees
  15. Escaping Tennenbaum's Theorem with More Truths
  16. ...and 3 more sections

Key Result

Theorem 1.1.7

The theories $\mathsf{PA}$ and $\mathsf{ZFfin}\mathord{+}\mathsf{TC}$ are definitionally equivalent.

Theorems & Definitions (82)

  • Definition 1.1.1: Definitional Extension
  • Definition 1.1.2: Definitional Equivalence
  • Definition 1.1.3
  • Remark 1.1.4
  • Definition 1.1.5
  • Remark 1.1.6
  • Theorem 1.1.7
  • Proof 1
  • Definition 1.2.1
  • Theorem 1.2.2
  • ...and 72 more