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Projective Games on the Reals

Juan P. Aguilera & Sandra Müller

Notre Dame Journal of Formal Logic 61 (4):573-589 (2020)

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Mice with finitely many Woodin cardinals from optimal determinacy hypotheses.Sandra Müller, Ralf Schindler & W. Hugh Woodin - 2020 - Journal of Mathematical Logic 20 (Supp01):1950013.details We prove the following result which is due to the third author. Let [Formula: see text]. If [Formula: see text] determinacy and [Formula: see text] determinacy both hold true and there is no [Formula: see text]-definable [Formula: see text]-sequence of pairwise distinct reals, then [Formula: see text] exists and is [Formula: see text]-iterable. The proof yields that [Formula: see text] determinacy implies that [Formula: see text] exists and is [Formula: see text]-iterable for all reals [Formula: see text]. A consequence is (...) Download Export citation Bookmark 6 citations
The consistency strength of long projective determinacy.Juan P. Aguilera & Sandra Müller - 2019 - Journal of Symbolic Logic 85 (1):338-366.details We determine the consistency strength of determinacy for projective games of length ω^2. Our main theorem is that $\Pi _{n + 1}^1$-determinacy for games of length ω^2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that M_n(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $A = R$ and the (...) Download Export citation Bookmark 2 citations
Projectively well-ordered inner models.J. R. Steel - 1995 - Annals of Pure and Applied Logic 74 (1):77-104.details Download Export citation Bookmark 29 citations
Deconstructing inner model theory.Ralf-Dieter Schindler, John Steel & Martin Zeman - 2002 - Journal of Symbolic Logic 67 (2):721-736.details Download Export citation Bookmark 17 citations

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