BI-INTERPRETATION IN WEAK SET THEORIES

Autor(es)

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

Afiliação do(s) autor(es)

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

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

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

• Tipo de documento: Artigo Científico
• Fonte: JOURNAL OF SYMBOLIC LOGIC; v. 86, n. 2, p. 609-634, JUN 2021.
• Citações Web of Science: 0

Resumo

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)

• Processo FAPESP: 17/21020-0 - Sobre as condições para uma comparação do comprometimento ontológico entre teorias
• Beneficiário: Alfredo Roque de Oliveira Freire Filho
• Modalidade de apoio: Bolsas no Exterior - Estágio de Pesquisa - Doutorado Direto