Journal of Symbolic Logic

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

Benjamin Claverie and Ralf Schindler

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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 ℛmax 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 “NSω₁ is precipitous” and strengthenings thereof. Along the way, we answer a question of Baumgartner and Taylor, [2, Question 6.11].

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J. Symbolic Logic, Volume 77, Issue 2 (2012), 475-498.

First available in Project Euclid: 4 April 2012

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Claverie, Benjamin; Schindler, Ralf. Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω₁. J. Symbolic Logic 77 (2012), no. 2, 475--498. doi:10.2178/jsl/1333566633.

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