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GAMES AND REFLECTION IN
The Journal of Symbolic Logic ( IF 0.6 ) Pub Date : 2020-07-21 , DOI: 10.1017/jsl.2020.20 J. P. AGUILERA
The Journal of Symbolic Logic ( IF 0.6 ) Pub Date : 2020-07-21 , DOI: 10.1017/jsl.2020.20 J. P. AGUILERA
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:
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。
更新日期:2020-07-21
中文翻译:
游戏与反思
我们描述了确定性