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Long games and σ-projective sets
Annals of Pure and Applied Logic ( IF 0.8 ) Pub Date : 2021-01-05 , DOI: 10.1016/j.apal.2020.102939
Juan P. Aguilera , Sandra Müller , Philipp Schlicht

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 <ω2 (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).



中文翻译:

长游戏和σ-投影集

我们证明了σ-实投影集的确定性的许多结果,即属于最小点类的那些实点,其中包含开放集并在补数,可数并集和投影下闭合。我们首先证明σ-投影确定性与某些类型的可变长度博弈的确定性之间的等价性<ω2(定理2.4)。然后,我们从最佳大基数假设(定理4.4)给出σ-射影集确定性的基本证明。最后,我们介绍如何将证明,得到的确定性的证明σ给定可数长度的-projective游戏和与在最小支付游戏σ含有射影套代数,从对应的假设(定理5.1,定理5.4 )。

更新日期:2021-01-12
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