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. The crucial ingredient of our proof is a version of the Schröder–Bernstein theorem under very weak conditions. We think this last result has some independent interest. We provide an…