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Choiceless large cardinals and set‐theoretic potentialism
Author(s) -
Cutolo Raffaella,
Hamkins Joel David
Publication year - 2022
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.202000026
Subject(s) - axiom , mathematics , modal , accessibility relation , possible world , universe , relation (database) , zermelo–fraenkel set theory , set theory , set (abstract data type) , discrete mathematics , modal logic , axiom of choice , mathematical economics , computer science , epistemology , philosophy , programming language , chemistry , physics , geometry , database , astrophysics , polymer chemistry
We define a potentialist system of ZF $\mathsf {ZF}$ ‐structures, i.e., a collection of possible worlds in the language of ZF $\mathsf {ZF}$ connected by a binary accessibility relation , achieving a potentialist account of the full background set‐theoretic universe V . The definition involves Berkeley cardinals, the strongest known large cardinal axioms, inconsistent with the Axiom of Choice. In fact, as background theory we assume just ZF $\mathsf {ZF}$ . It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theoryS 4 . 2 $\mathsf {S4.2}$ . Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theoryS 5 $\mathsf {S5}$ , both for assertions in the language of ZF $\mathsf {ZF}$ and for assertions in the full potentialist language.