实数上的投影博弈
Projective Games on the Reals
摘要：
令$M^\sharp_n(\mathbb{R})$表示具有$n$ Woodin基数并包含所有实数的最小活动可迭代扩展模型，如果它存在的话，在这种情况下，我们用$M_n(\mathbb{R})$表示通过多次迭代$M_n(\mathbb{R})$类的最高测度而获得的类大小模型。在实数上的投影博弈确定的前提下，刻画了由$\mathbb{R}$ over $M_n(\mathbb{R})$可定义为$Σ_1$的实数集：（1）对于偶数$n$，$Σ_1^{M_n(\mathbb{R})} = \Game^\mathbb{R}Π^1_{n+1}$ ;（2）对于奇数$n$，$Σ_1^{M_n(\mathbb{R})} = \Game^\mathbb{R}Σ^1_{n+1}$ .这推广了Martin和Steel关于$L(\mathbb{R})$的定理，即，$n=0$案作为证明的结果，我们看到，所有在$\mathbb{R}$中移动的投射博弈的确定性等价于$M^\sharp_n(\mathbb{R})$对所有$n\in\mathbb{N}$存在的陈述，并且所有在$\mathbb{N}$中移动的长度为$ω^2$的投射博弈的确定性等价于$M^\sharp_n(\mathbb{R})$对所有$n\in\mathbb{N}$存在并满足$\mathsf{AD}$的陈述。
英文摘要：
Let $M^\sharp_n(\mathbb{R})$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n(\mathbb{R})$ the class-sized model obtained by iterating the topmost measure of $M_n(\mathbb{R})$ class-many times. We characterize the sets of reals which are $Σ_1$-definable from $\mathbb{R}$ over $M_n(\mathbb{R})$, under the assumption that projective games on reals are determined: (1) for even $n$, $Σ_1^{M_n(\mathbb{R})} = \Game^\mathbb{R}Π^1_{n+1}$; (2) for odd $n$, $Σ_1^{M_n(\mathbb{R})} = \Game^\mathbb{R}Σ^1_{n+1}$. This generalizes a theorem of Martin and Steel for $L(\mathbb{R})$, i.e., the case $n=0$. As consequences of the proof, we see that determinacy of all projective games with moves in $\mathbb{R}$ is equivalent to the statement that $M^\sharp_n(\mathbb{R})$ exists for all $n\in\mathbb{N}$, and that determinacy of all projective games of length $ω^2$ with moves in $\mathbb{N}$ is equivalent to the statement that $M^\sharp_n(\mathbb{R})$ exists and satisfies $\mathsf{AD}$ for all $n\in\mathbb{N}$.
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