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  1. Calibrating determinacy strength in levels of the borel hierarchy.Sherwood Hachtman - 2017 - Journal of Symbolic Logic 82 (2):510-548.
    We analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form$\Sigma _{1 + \alpha + 3}^0 $, forα<ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to requireα+ 1 iterations of the Power Set Axiom, but we ask what additional ambient set theory is strictly necessary. To this end, we isolate a family of weak reflection principles, Π1-RAPα, whose consistency strength corresponds exactly to the logical strength of${\rm{\Sigma }}_{1 + \alpha + (...)
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  • Weak systems of determinacy and arithmetical quasi-inductive definitions.P. D. Welch - 2011 - Journal of Symbolic Logic 76 (2):418 - 436.
    We locate winning strategies for various ${\mathrm{\Sigma }}_{3}^{0}$ -games in the L-hierarchy in order to prove the following: Theorem 1. KP+Σ₂-Comprehension $\vdash \exists \alpha L_{\alpha}\ models"\Sigma _{2}-{\bf KP}+\Sigma _{3}^{0}-\text{Determinacy}."$ Alternatively: ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ "there is a β-model of ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17 em}}{\mathrm{\Sigma }}_{3}^{0}$ -Determinacy." The implication is not reversible. (The antecedent here may be replaced with ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\mathrm{\Pi }}_{3}^{1}\right)-{\mathrm{C}\mathrm{A}}_{0}:\text{\hspace{0.17em}}{\mathrm{\Pi }}_{3}^{1}$ instances of Comprehension with only ${\mathrm{\Pi }}_{3}^{1}$ -lightface definable parameters—or even weaker theories.) Theorem 2. KP +Δ₂-Comprehension +Σ₂-Replacement + ${\mathrm{\Sigma }}_{3}^{0}\phantom{\rule{0ex}{0ex}}$ -Determinacy. (Here AQI (...)
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  • Admissible Sets and Structures. An Approach to Definability Theory.Mark Nadel - 1978 - Journal of Symbolic Logic 43 (1):139-144.
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  • Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
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  • Subsystems of Second Order Arithmetic.Stephen G. Simpson - 1999 - Studia Logica 77 (1):129-129.
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  • Higher set theory and mathematical practice.Harvey M. Friedman - 1971 - Annals of Mathematical Logic 2 (3):325.
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  • Set Theory.Keith J. Devlin - 1981 - Journal of Symbolic Logic 46 (4):876-877.
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  • Weak axioms of determinacy and subsystems of analysis II.Kazuyuki Tanaka - 1991 - Annals of Pure and Applied Logic 52 (1-2):181-193.
    In [10], we have shown that the statement that all ∑ 1 1 partitions are Ramsey is deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition,but the reversal needs П 1 1 - CA 0 rather than ATR 0 . By contrast, we show in this paper that the statement that all ∑ 0 2 games are determinate is also deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition, but the (...)
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  • Admissible Sets and Structures.Jon Barwise - 1978 - Studia Logica 37 (3):297-299.
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  • Transfinite recursion in higher reverse mathematics.Noah Schweber - 2015 - Journal of Symbolic Logic 80 (3):940-969.
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  • The fine structure of the constructible hierarchy.R. Björn Jensen - 1972 - Annals of Mathematical Logic 4 (3):229.
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