This will be a talk for the workshop in Set Theory at the Mathematisches Forschungsinstitute Oberwolfach, April 5-11, 2020.

**Abstract:** Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above. Nevertheless, for natural weaker set theories, we prove, including 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 $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending 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. This is joint work with Alfredo Roque Freire.

Bi-interpretation in weak set theories

- J. D. Hamkins and A. R. Freire, “Bi-interpretation in weak set theories,” Mathematics arXiv, 2020.
`@ARTICLE{HamkinsFreire:Bi-interpretation-in-weak-set-theories, author = {Joel David Hamkins and Alfredo Roque Freire}, title = {Bi-interpretation in weak set theories}, journal = {Mathematics arXiv}, year = {2020}, volume = {}, number = {}, pages = {}, month = {}, note = {}, abstract = {}, keywords = {}, source = {}, doi = {}, url = {http://jdh.hamkins.org/bi-interpretation-in-weak-set-theories}, eprint = {2001.05262}, archivePrefix = {arXiv}, primaryClass = {math.LO}, }`

- J. D. Hamkins and A. R. Freire, “Bi-interpretation in weak set theories,” Mathematics arXiv, 2020.