The Σ2-Potentialist Principle

Abstract

We settle a question of Woodin motivated by the philosophy of potentialism in set theory. A sentence in the language of set theory is locally verifiable if it asserts the existence of a level $V_{\alpha }$ of the cumulative hierarchy of sets with some first-order property; this is equivalent to being ${\Sigma }_2$ in the Lévy hierarchy. A sentence is $V_{\alpha }$-satisfiable if it can be forced without changing $V_{\alpha }$, and V-satisfiable if it is $V_{\alpha }$-satisfiable for all ordinals α. The ${\Sigma }_2$-Potentialist Principle, introduced by Woodin, asserts that every V-satisfiable locally verifiable sentence is true. We show in Theorem 6.2 that the ${\Sigma }_2$-Potentialist Principle is consistent relative to a supercompact cardinal. We accomplish this by generalizing Gitik's method of iterating distributive forcings by embedding them into Príkry-type forcings [6, Section 6.4]; our generalization, Theorem 5.2, works for forcings that add no bounded subsets to a strongly compact cardinal, which requires a completely different proof. Finally, using the concept of mutual stationarity, we show in Theorem 7.5 that the ${\Sigma }_2$-Potentialist Principle implies the consistency of a Woodin cardinal.³