Journal of Symbolic Logic 86 (2):609-634 (2021)

Authors
Joel David Hamkins
Oxford University
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 $\mathrm {ZFC}^{-}$ without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of $\mathrm {ZFC}^{-}$ 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 $\mathrm {ZFC}^{-}$. 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.
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DOI 10.1017/jsl.2020.72
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References found in this work BETA

The Set-Theoretic Multiverse.Joel David Hamkins - 2012 - Review of Symbolic Logic 5 (3):416-449.
Set-Theoretic Geology.Gunter Fuchs, Joel David Hamkins & Jonas Reitz - 2015 - Annals of Pure and Applied Logic 166 (4):464-501.
A Natural Model of the Multiverse Axioms.Victoria Gitman & Joel David Hamkins - 2010 - Notre Dame Journal of Formal Logic 51 (4):475-484.

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Citations of this work BETA

Reflection Principles and Second-Order Choice Principles with Urelements.Bokai Yao - 2022 - Annals of Pure and Applied Logic 173 (4):103073.

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