Choiceless large cardinals and set-theoretic potentialism

TL;DR

This paper constructs a potentialist framework for the set-theoretic universe using Berkeley cardinals, showing that modal validities align with S4.2 and S5 theories, even under ZF without Choice.

Contribution

It introduces a potentialist system of ZF-structures based on Berkeley cardinals, providing a modal logic characterization of the set-theoretic universe.

Findings

Modal validities are exactly those in S4.2.

Worlds satisfying the maximality principle validate S5.

The framework works within ZF, without assuming the Axiom of Choice.

Abstract

We define a potentialist system of ZF-structures, that is, a collection of possible worlds in the language of 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. 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 ZF and for assertions in the full potentialist language.