The Σ2-Potentialist Principle

Woodin’s Σ₂-Potentialist Principle roughly asserts that every locally verifiable set-theoretic statement that is consistent in a strong sense is true. Woodin asked whether this principle is itself consistent. The motivation is that a negative answer would give a mathematical argument against a certain form of set-theoretic potentialism. Using techniques from Gitik’s theory of iterated Prikry-type forcing, we give a positive answer to Woodin’s question starting with a supercompact cardinal. Appealing to recent results of Adolf–Ben Nera–Zeman on the strength of mutual stationarity, we give a consistency strength lower bound on the principle of one Woodin cardinal. This is joint work with Omer Ben Neria and Eyal Kaplan.