When Bi-interpretability implies Synonymy

TL;DR

This paper investigates when bi-interpretability of sequential theories implies their synonymy, establishing conditions under which these notions coincide and providing examples and a Schröder-Bernstein theorem variant.

Contribution

It proves that bi-interpretability implies synonymy for sequential theories with identity-preserving, one-dimensional interpretations, and introduces a novel Schröder-Bernstein theorem under weak conditions.

Findings

Bi-interpretability implies synonymy under specific conditions.

Counterexamples show the necessity of these conditions.

A new Schröder-Bernstein theorem with weak assumptions.

Abstract

Two salient notions of sameness of theories are synonymy, also known as definitional equivalence, and bi-interpretability. Of these two definitional equivalence is the strictest notion. In which cases can we infer synonymy from bi-interpretability? We study this question for the case of sequential theories. Our result is as follows. Suppose that two sequential theories are bi-interpretable and that the interpretations involved in the bi-interpretation are one-dimensional and identity preserving. Then, the theories are synonymous. The crucial ingredient of our proof is a version of the Schr\"oder-Bernstein theorem under very weak conditions. We think this last result has some independent interest. We provide an example to show that this result is optimal. There are two finitely axiomatized sequential theories that are bi-interpretable but not synonymous, where precisely one of the interpretations involved in the bi-interpretation is not identity preserving.