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THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY

Published online by Cambridge University Press:  18 November 2019

JUAN P. AGUILERA  and
SANDRA MÜLLER
JUAN P. AGUILERA
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF GHENT KRIJGSLAAN 281-S8, B9000 GHENT BELGIUM and INSTITUTE OF DISCRETE MATHEMATICS AND GEOMETRY VIENNA UNIVERSITY OF TECHNOLOGY WIEDNER HAUPTSTRAßE 8-10, 1040 VIENNA AUSTRIA E-mail:aguilera@logic.at
SANDRA MÜLLER
Affiliation:
INSTITUT FÜR MATHEMATIK, UZA 1 UNIVERSITÄT WIEN AUGASSE 2-6, 1090 WIEN AUSTRIA E-mail:mueller.sandra@univie.ac.at
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Abstract

We determine the consistency strength of determinacy for projective games of length ω2. Our main theorem is that $\Pi _{n + 1}^1 $-determinacy for games of length ω2 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 Mn (A), the canonical inner model for n Woodin cardinals constructed over A, satisfies $$A = R$$ and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.

We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω2 with payoff in $^R R\Pi _1^1 $ or with σ-projective payoff.

Keywords

03E4503E6003E1503E55infinite gamedeterminacyinner model theorylarge cardinallong gamemouse
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 338 - 366
Copyright
Copyright © The Association for Symbolic Logic 2019 

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