The determined property of Baire in reverse math
| Autor: | ERIC P. ASTOR, DAMIR DZHAFAROV, ANTONIO MONTALBÁN, REED SOLOMON, LINDA BROWN WESTRICK |
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| Rok vydání: | 2018 |
| Předmět: |
Discrete mathematics
Code (set theory) Reduction (recursion theory) Logic 010102 general mathematics Mathematics - Logic 0102 computer and information sciences 01 natural sciences Philosophy Mathematics::Logic 010201 computation theory & mathematics FOS: Mathematics Reverse mathematics 03B30 Property of Baire 0101 mathematics Borel set Logic (math.LO) Mathematics |
| DOI: | 10.48550/arxiv.1809.03940 |
| Popis: | 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 $Z$. Comment: Greatly expanded introduction as requested by referee |
| Databáze: | OpenAIRE |
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