A theory satisfying a strong version of Tennenbaum's theorem

TL;DR

This paper constructs a consistent, computably enumerable theory with the property that no definitionally equivalent theory admits a computable model, using mutual algebraicity as a key tool.

Contribution

It introduces a theory satisfying a strong version of Tennenbaum's theorem, demonstrating limitations on computable models for certain theories.

Findings

Existence of a c.e. theory with no computable models under definitional equivalence

Application of mutual algebraicity in model theory

Answer to Pakhomov's question about computable models

Abstract

We answer a question of Pakhomov by showing that there is a consistent, c.e. theory $T$ such that no theory which is definitionally equivalent to $T$ has a computable model. A key tool in our proof is the model-theoretic notion of mutual algebraicity.