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

Published online by Cambridge University Press:  24 February 2017

Benjamin Claverie  and
Ralf Schindler
Benjamin Claverie
Affiliation:
Institut Für Mathematische Logik Und Grundlagenforschung, Universität MünsterEinsteinstr. 62, 48149 Münster, Germany, E-mail: [email protected]:, [email protected]
Ralf Schindler
Affiliation:
Institut Für Mathematische Logik Und Grundlagenforschung, Universität MünsterEinsteinstr. 62, 48149 Münster, Germany, E-mail: [email protected]:, [email protected]
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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 ℙmax 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].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 475 - 498
Copyright
Copyright © Association for Symbolic Logic 2012

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