Bi-interpretation in weak set theories - Freire, AR; Hamkins, JD (Cambridge University Press, 2020)
In contrast to the robust mutual interpretability phenomenon in set theory, Ali Enayat proved that bi-interpretation is absent: distinct theories extending ZF are never bi-interpretable and models of ZF are bi-interpretable only when they are isomorphic. Nevertheless, for natural weaker set theories, we prove, including Zermelo-Fraenkel set theory ZFC− without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC− that are bi-interpretable, but not isomorphic—even hHω, ∈i and hHω2, ∈i can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC−. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails.