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- Title
THE CONSISTENCY STRENGTH OF LONG PROJECTIVE DETERMINACY.
- Authors
AGUILERA, JUAN P.; MÜLLER, SANDRA
- Abstract
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 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 ω 2 with payoff in $^R R\Pi _1^1 $ or with σ -projective payoff.
- Subjects
CARDINAL numbers; SET theory; MODEL theory; NONCOOPERATIVE games (Mathematics); AXIOMS
- Publication
Journal of Symbolic Logic, 2020, Vol 85, Issue 1, p338
- ISSN
0022-4812
- Publication type
Article
- DOI
10.1017/jsl.2019.78