GAMES AND REFLECTION IN

Pub Date : 2020-07-21 DOI:10.1017/jsl.2020.20

J. P. Aguilera

引用次数: 1

Abstract

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.

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游戏与反思

我们根据短博弈的确定性断言来表征长度为$\omega ^2$的$F_\sigma $博弈的确定性。具体地说，我们证明了长度为$\omega ^2$的$F_\sigma $对策是确定的，当且仅当存在一个包含$\mathbb {R}$并反映关于下一个可容许集的$\Pi _1$事实的${\mathsf {KP}}+{\mathsf {AD}}$传递模型。因此，根据${\mathsf {KP}} + {\mathsf {DC}} + ``\mathbb {R}$存在的基本理论，“长度为$\omega ^2$的$F_\sigma $游戏的确定性比${\mathsf {AD}}$强，但比${\mathsf {AD}} + \Sigma _1$ -分离弱。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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