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

 @ARTICLE{AMS21,
   AUTHOR= {J. Aguilera and S. Müller and P. Schlicht},
   TITLE= {Long games and {$\sigma$-projective} sets},
   JOURNAL = {Annals of Pure and Applied Logic},
   VOLUME = {172},
   ISSUE = {4},
   YEAR = {2021},
   PAGES = {102939},
   DOI = {10.1016/j.apal.2020.102939},
   EPRINT = {2011.04947}}

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).