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Choiceless large cardinals and set-theoretic potentialism

We define a potentialist system of ZF-structures, that is, a collection of possible worlds in the language of 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. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the modal theory S4.2. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory S5, both for assertions in the language of ZF and for assertions in the full potentialist language.