# Bi-interpretation of weak set theories, Oberwolfach, April 2020

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

Note: the conference has been cancelled due to concerns over the Coronavirus-19. (Meanwhile, I have given the talk for the Oxford Set Theory Seminar — see below.)

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.

Since the Oberwolfach meeting had been canceled, I gave the talk for the Oxford Set Theory Seminar on 20 May 2020.

• A. R. Freire and J. D. Hamkins, “Bi-interpretation in weak set theories,” Mathematics arXiv, 2020. (Under review)
@ARTICLE{FreireHamkins:Bi-interpretation-in-weak-set-theories,
author = {Alfredo Roque Freire and Joel David Hamkins},
title = {Bi-interpretation in weak set theories},
journal = {Mathematics arXiv},
year = {2020},
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note = {Under review},
abstract = {},
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url = {http://jdh.hamkins.org/bi-interpretation-in-weak-set-theories},
eprint = {2001.05262},
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primaryClass = {math.LO},
}

# The Ground Axiom

• J. D. Hamkins, “The Ground Axiom,” Mathematisches Forschungsinstitut Oberwolfach Report, vol. 55, pp. 3160-3162, 2005.
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AUTHOR = "Joel David Hamkins",
TITLE = "The {Ground Axiom}",
JOURNAL = "Mathematisches Forschungsinstitut Oberwolfach Report",
YEAR = "2005",
volume = "55",
number = "",
pages = "3160--3162",
month = "",
note = "",
abstract = "",
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eprint = {1607.00723},
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}

This is an extended abstract for a talk I gave at the 2005 Workshop in Set Theory at the Mathematisches Forschungsinstitut Oberwolfach.