  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 $\text{ZFC}^-$ without power set and
Zermelo set theory Z, there are nontrivial instances of bi-interpretation.
Specifically, there are well-founded models of $\text{ZFC}^-$ that are
bi-interpretable, but not isomorphic---even $\langle H_{\omega_1},\in\rangle$
and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable---and there are
distinct bi-interpretable theories extending $\text{ZFC}^-$. 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.
