首页/论文/2001.05262

arXiv 2001.052622020-01math.LO

Bi-interpretation in weak set theories

Alfredo Roque Freire, Joel David Hamkins

备注：: 25 pages. Commentary can be made about this article on the second author's blog at http://jdh.hamkins.org/bi-interpretation-in-weak-set-theories. Version 2 corrects the author order to alphabetical and makes some minor inclusions regarding the difference between solidity and semantic tightness and the mutual interpretation of a model of set theory with the theory of its forcing extensions

英文摘要：

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 $\langle H_{ω_1},\in\rangle$ and $\langle H_{ω_2},\in\rangle$ 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.

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