When Bi-Interpretability Implies Synonymy

Abstract

Two salient notions of sameness of theories are synonymy, aka 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. 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. The crucial ingredient of our proof is a version of the Schröder-Bernstein theorem under very weak conditions. We think this result has some independent interest.