Long Games and $\sigma$-Projective Sets

TL;DR

This paper explores the determinacy of $\sigma$-projective sets of reals, establishing equivalences with certain game classes, providing elementary proofs from large cardinal hypotheses, and generalizing to games of various lengths and payoffs.

Contribution

It introduces new equivalences for $\sigma$-projective determinacy, offers elementary proofs from large cardinals, and extends results to games of different lengths and payoff classes.

Findings

Equivalence between $\sigma$-projective determinacy and certain variable-length game determinacies

Elementary proof of $\sigma$-projective determinacy from large cardinal hypotheses

Generalization to games of various lengths and payoff classes

Abstract

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 (Theorems 5.1 and 5.4).