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arXiv · 1809.03940

The determined property of Baire in reverse math

Eric P. Astor·Damir Dzhafarov·Antonio Montalbán·Reed Solomon·Linda Brown Westrick
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Abstract

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

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Eric P. Astor, Damir Dzhafarov, Antonio Montalbán, Reed Solomon, Linda Brown Westrick. 2020-07-05. The determined property of Baire in reverse math. https://doi.org/10.1017/jsl.2019.64

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