Long games and sigma-projective sets

(with J. Aguilera and P. Schlicht)

Annals of Pure and Applied Logic. Volume 172, Issue 4, April 2021. 102939.
DOI: 10.1016/j.apal.2020.102939. PDF. arXiv. Bibtex.

We prove a number of results on the determinacy of $\sigma$-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 $\sigma$-projective determinacy and the determinacy of certain classes of games of variable length ${<}\omega^2$ (Theorem 2.4). We then give an elementary proof of the determinacy of $\sigma$-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 $\sigma$-projective games of a given countable length and of games with payoff in the smallest $\sigma$-algebra containing the projective sets, from corresponding assumptions (Theorem 5.1, Theorem 5.4).