Long games and σ-projective sets

Annals of Pure and Applied Logic 172 (4):102939 (2021)
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Abstract

We prove a number of results on the determinacy of σ-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between σ-projective determinacy and the determinacy of certain classes of games of variable length <ω^2 (Theorem 2.4). We then give an elementary proof of the determinacy of σ-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of the determinacy of σ-projective games of a given countable length and of games with payoff in the smallest σ-algebra containing the projective sets, from corresponding assumptions (Theorem 5.1, Theorem 5.4).

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Author Profiles

Sandra Eleonore Müller
Ludwig Maximilians Universität, München

References found in this work

Optimal proofs of determinacy.Itay Neeman - 1995 - Bulletin of Symbolic Logic 1 (3):327-339.
Descriptive Set Theory.Yiannis Nicholas Moschovakis - 1982 - Studia Logica 41 (4):429-430.
Deconstructing inner model theory.Ralf-Dieter Schindler, John Steel & Martin Zeman - 2002 - Journal of Symbolic Logic 67 (2):721-736.

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