BI-INTERPRETATION IN WEAK SET THEORIES

Author(s)

Freire, Alfredo Roque ^[1] ; Hamkins, Joel David ^{[2, 3]}

Affiliation

^[1] Univ Brasilia, Philosophy Dept, Brasilia, DF - Brazil

^[2] Univ Coll, Philosophy, Oxford - England

^[3] Univ Oxford, Log, Fac Philosophy, Dept Math, Oxford - England

• Document type: Journal article
• Source: JOURNAL OF SYMBOLIC LOGIC; v. 86, n. 2, p. 609-634, JUN 2021.
• Web of Science Citations: 0

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 < H-omega 1, is an element of > and < H-omega 2, is an element of > 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. (AU)

• FAPESP's process: 17/21020-0 - On the conditions for a comparison of ontological commitment between theories
• Grantee: Alfredo Roque de Oliveira Freire Filho
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate (Direct)