Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω₁

Abstract

If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at ℵ₂ with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC—model. This yields that if Woodin's ℛ_max axiom (*) holds, then BPFA implies that V is closed under the “Woodin-in-the-next-ZFC—model” operator. We also discuss stronger Mouse Reflection principles which we show to follow from strengthenings of BPFA, and we discuss the theory BPFA plus “NS_ω₁ is precipitous” and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor, [2, Question 6.11].