The Σ₂-Potentialist Principle

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_α of the cumulative hierarchy of sets with some first-order property; this is equivalent to being Σ₂ in the Lévy hierarchy. A sentence is V_α-satisfiable if it can be forced without changing V_α, and V-satisfiable if it is V_α-satisfiable for all ordinals α. The Σ₂-Potentialist Principle, introduced by Woodin, asserts that every V-satisfiable locally verifiable sentence is true. We show in Theorem 6.2 that the Σ₂-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 Σ₂-Potentialist Principle implies the consistency of a Woodin cardinal.³