How to escape Tennenbaum's theorem

TL;DR

This paper demonstrates the construction of non-standard computable models for theories equivalent to Peano arithmetic and ZF set theory, challenging traditional limitations on computability within these frameworks.

Contribution

It introduces a method to build computable models for theories definitionally equivalent to PA and ZF, addressing Tennenbaum's theorem.

Findings

Constructed a non-standard computable model of PA

Built a computable model of ZF set theory

Challenged assumptions of Tennenbaum's theorem

Abstract

We construct a theory definitionally equivalent to first-order Peano arithmetic PA and a non-standard computable model of this theory. The same technique allows us to construct a theory definitionally equivalent to Zermelo-Fraenkel set theory ZF that has a computable model.