# Bi-interpretation in set theory, Bristol, February 2020

This will be a talk for the Logic and Set Theory seminar at the University of Bristol, on 25 February, 2020.

Abstract: 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 $\langle H_{\omega_1},\in\rangle$ 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},
}