Choiceless large cardinals and set‐theoretic potentialism

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.