Choiceless large cardinals and set-theoretic potentialism

We define a potentialist system of $\mathrm{ZF}$-structures, i.e., a collection of possible worlds in the language of $\mathrm{ZF}$ 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 $\mathrm{ZF}$. 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 $\mathrm{ZF}$ and for assertions in the full potentialist language.