THE DETERMINED PROPERTY OF BAIRE IN REVERSE MATH

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...

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Bibliographic Details
Published in:The Journal of symbolic logic 2020-03, Vol.85 (1), p.166-198
Main Authors: ASTOR, ERIC P., DZHAFAROV, DAMIR, MONTALBÁN, ANTONIO, SOLOMON, REED, WESTRICK, LINDA BROWN
Format: Article
Language:English
Subjects:
Logic
Mathematics
Principles
Theorems
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Summary: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 .
ISSN:0022-4812
1943-5886
DOI:10.1017/jsl.2019.64