We study games of length ω² with moves in ℕ and Borel payoff. These are, e.g., games in which two players alternate turns playing digits to produce a real number in [0, 1] infinitely many times, after which the winner is decided in terms of the sequence belonging to a Borel set in the product space [0,1]^ℕ.

The main theorem is that Borel games of length ω² are determined if, and only if, for every countable ordinal α, there is a fine-structural, countably iterable model of Zermelo set theory with α-many iterated powersets above a limit of Woodin cardinals.

中文翻译：

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长波雷尔游戏

我们研究了长度为ω ^{2 的}博弈，其动作为 ℕ 和 Borel 收益。例如，在这些游戏中，两个玩家交替轮流玩数字以无限多次产生 [0, 1] 中的实数，然后根据属于乘积空间中的 Borel 集的序列决定获胜者 [0, 1]。 0,1] ^ℕ .

主要定理是长度为ω ^{2 的}博雷尔博弈是确定的，当且仅当对于每个可数序数α，存在一个精细结构的、可数迭代的 Zermelo 集合理论模型，其中α -许多迭代幂集高于 Woodin 的极限红雀。