论文标题
长游戏和$σ$ -Projective套件
Long Games and $σ$-Projective Sets
论文作者
论文摘要
我们证明了$σ$ -Projective roeals集的确定性的许多结果，即属于包含开放式集合的最小点级的元素，并在补充，可计数的工会和预测下关闭。我们首先证明了$σ$ -Projective的确定性与某些类别的可变长度游戏游戏的确定性之间的等价性，$ {<}ω^2 $（Theorem 2.4）。然后，我们给出一个基本证明，证明了最佳大型假设（定理4.4）的$σ$ -Projective集的决定性。最后，我们展示了如何概括证明以获取给定可计数长度的$σ$ - 标记游戏的确定性证明，并且从相应的假设（定理5.1和5.4）中获得了包含投影集的最小$σ$ -Algebra的游戏。
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).
