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THE DETERMINED PROPERTY OF BAIRE IN REVERSE MATH
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- Journal:
- The Journal of Symbolic Logic / Volume 85 / Issue 1 / March 2020
- Published online by Cambridge University Press:
- 10 September 2019, pp. 166-198
- Print publication:
- March 2020
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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.
Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set
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- Journal:
- Canadian Journal of Mathematics / Volume 57 / Issue 6 / 01 December 2005
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1139-1154
- Print publication:
- 01 December 2005
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We prove that it is relatively consistent with
$ZFC$ that in any perfect Polish space, for every nonmeager set 
$A$ there exists a nowhere dense Cantor set 
$C$ such that 
$A\,\cap \,C$ is nonmeager in $C$. We also examine variants of this result and establish a measure theoretic analog.
