The determined property of Baire in reverse math

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 $\omega$-model of $DPB$ must be closed under hyperarithmetic reduction, but $DPB$ is not a theory of hyperarithmetic analysis. We show that whenever $M\subseteq 2^\omega$ is the second-order part of an $\omega$-model of $DPB$, then for every $Z \in M$, there is a $G \in M$ such that $G$ is $\Delta^1_1$-generic relative to $Z$.