We define a potentialist system of ‐structures, i.e., a collection of possible worlds in the language of 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. It turns out that the propositional modal assertions which are valid at every world of our system are exactly those in the (...) modal theory. Moreover, we characterize the worlds satisfying the potentialist maximality principle, and thus the modal theory, both for assertions in the language of and for assertions in the full potentialist language. (shrink)

For a given inner model N of ZFC, one can consider the relativized version of Berkeley cardinals in the context of ZFC, and ask if there can exist an “N-Berkeley cardinal.” In this article we provide a positive answer to this question. Indeed, under the assumption of a supercompact cardinal $\delta $, we show that there exists a ZFC inner model N such that there is a cardinal which is N-Berkeley, even in a strong sense. Further, the involved model N (...) is a weak extender model of $\delta $ is supercompact. Finally, we prove that the strong version of N-Berkeley cardinals turns out to be inconsistent whenever N satisfies closure under $\omega $ -sequences. (shrink)

This paper is concerned with the possible values of the cofinality of the least Berkeley cardinal. Berkeley cardinals are very large cardinal axioms incompatible with the Axiom of Choice, and the interest in the cofinality of the least Berkeley arises from a result in [1], showing it is connected with the failure of. In fact, by a theorem of Bagaria, Koellner and Woodin, if γ is the cofinality of the least Berkeley cardinal then γ‐ fails. We shall prove that this (...) result is optimal for or. In particular, it will follow that the cofinality of the least Berkeley is independent of. (shrink)