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

Choiceless large cardinals and set-theoretic potentialism