We define the notion of a completely determined Borel code in reverse mathematics, and consider the principle $CD - PB$, which states that every completely determined Borel set has the property of Baire. We show that this principle is strictly weaker than $AT{R_0}$. Any ω-model of $CD - PB$ must be closed under hyperarithmetic reduction, but $CD - PB$ is not a theory of hyperarithmetic analysis. We show that whenever $M \subseteq {2^\omega }$ is the second-order part of an ω-model of $CD - PB$, then for every $Z \in M$, there is a $G \in M$ such that G is ${\rm{\Delta }}_1^1$-generic relative to Z.

中文翻译：

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反向数学中 BAIRE 的确定性质

我们在逆向数学中定义了完全确定的Borel码的概念，并考虑了原理$CD-PB$，它表明每个完全确定的 Borel 集都具有 Baire 的性质。我们证明了这个原则严格地弱于$AT{R_0}$. 任何ω-模型$CD-PB$必须在超算术减少下关闭，但$CD-PB$不是超算术分析的理论。我们表明，无论何时$M \subseteq {2^\omega }$是一个的二阶部分ω-模型$CD-PB$，那么对于每个$Z \in M$，有一个$G \in M$这样G是${\rm{\Delta }}_1^1$-泛型相对于Z.