Long Games and $σ$-Projective Sets

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 (Theorems 5.1 and 5.4).

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