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THE DETERMINED PROPERTY OF BAIRE IN REVERSE MATH

Published online by Cambridge University Press:  10 September 2019

ERIC P. ASTOR ,
DAMIR DZHAFAROV ,
ANTONIO MONTALBÁN ,
REED SOLOMON  and
LINDA BROWN WESTRICK
ERIC P. ASTOR
Affiliation:
GOOGLE LLC, 111 8TH AVE. NEW YORK, NY10011, USAE-mail:eric.astor@gmail.com
DAMIR DZHAFAROV
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CONNECTICUT STORRS, CONNECTICUT, USA E-mail:damir.dzhafarov@uconn.edu
ANTONIO MONTALBÁN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY CALIFORNIA-BERKELEY BERKELEY, CALIFORNIA, USA E-mail:antonio@math.berkeley.edu
REED SOLOMON
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CONNECTICUT STORRS, CONNECTICUT, USA E-mail:solomon@math.uconn.edu
LINDA BROWN WESTRICK
Affiliation:
DEPARTMENT OF MATHEMATICS PENN STATE UNIVERSITY UNIVERSITY PARK, PENNSYLVANIA, USA E-mail:westrick@psu.edu
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Abstract

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.

Keywords

03F35reverse mathematicsBorel setsproperty of Bairedual Ramsey theoremtheories of hyperarithmetic analysis

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Type
Articles
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The Journal of Symbolic Logic , Volume 85 , Issue 1 , March 2020 , pp. 166 - 198
Copyright
Copyright © The Association for Symbolic Logic 2019 

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