The modal logic of set-theoretic potentialism and the potentialist maximality principles

TL;DR

This paper investigates the modal logic principles underlying various forms of set-theoretic potentialism, establishing precise bounds for their modal validities and exploring the implications of potentialist maximality principles.

Contribution

It develops a general model-theoretic framework for analyzing set-theoretic potentialism, identifying modal bounds and maximality principles across multiple potentialist conceptions.

Findings

Lower bounds for modal validities are typically S4.2 or S4.3.

Upper bounds are S5, which are shown to be optimal.

S5 validity corresponds to potentialist maximality principles.

Abstract

We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and L\"owe, including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_\beta$); Grothendieck-Zermelo potentialism (true in all larger $V_\kappa$ for inaccessible cardinals $\kappa$); transitive-set potentialism (true in all larger transitive sets); forcing potentialism (true in all forcing extensions); countable-transitive-model potentialism (true in all larger countable transitive models of ZFC); countable-model potentialism (true in all larger countable models of ZFC); and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.