We characterize the determinacy of $F_\sigma $ games of length $\omega ^2$ in terms of determinacy assertions for short games. Specifically, we show that $F_\sigma $ games of length $\omega ^2$ are determined if, and only if, there is a transitive model of ${\mathsf {KP}}+{\mathsf {AD}}$ containing $\mathbb {R}$ and reflecting $\Pi _1$ facts about the next admissible set.As a consequence, one obtains that, over the base theory ${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$ exists,” determinacy for $F_\sigma $ games of length $\omega ^2$ is stronger than ${\mathsf {AD}}$ , but weaker than ${\mathsf {AD}} + \Sigma _1$ -separation.

中文翻译：

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游戏与反思

我们描述了确定性$F_\sigma $长度游戏$\欧米茄^2$就短游戏的确定性断言而言。具体来说，我们表明$F_\sigma $长度游戏$\欧米茄^2$当且仅当存在一个传递模型${\mathsf {KP}}+{\mathsf {AD}}$包含$\mathbb {R}$并反映$\Pi _1$关于下一个可接纳集的事实。因此，我们得到了，在基础理论之上${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$存在，“的确定性$F_\sigma $长度游戏$\欧米茄^2$强于${\mathsf {AD}}$, 但弱于${\mathsf {AD}} + \Sigma _1$-分离。