Choiceless large cardinals and set‐theoretic potentialism

We define a potentialist system of ZF-structures, that is, collection possible worlds in the language ZF connected by binary accessibility relation, achieving account full background set-theoretic universe $V$. The definition involves Berkeley cardinals, strongest known large cardinal axioms, inconsistent with Axiom Choice. In fact, as theory we assume just ZF. It turns out propositional modal assertions which are valid at every world our exactly those S4.2. Moreover, characterize satisfying maximality principle, and thus S5, both for language.