The determined property of Baire in reverse math

TL;DR

This paper introduces the principle DPB in reverse mathematics, showing it is weaker than ATR and exploring its implications for models and genericity, thus advancing understanding of Baire property in this context.

Contribution

It defines the notion of determined Borel codes and the DPB principle, establishing its position relative to ATR and analyzing model-theoretic properties.

Findings

DPB is strictly weaker than ATR.

Models of DPB are closed under hyperarithmetic reduction.

Models of DPB contain $ ext{Δ}^1_1$-generics relative to their elements.

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 $Z$.