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CALIBRATING DETERMINACY STRENGTH IN LEVELS OF THE BOREL HIERARCHY
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- Journal:
- The Journal of Symbolic Logic / Volume 82 / Issue 2 / June 2017
- Published online by Cambridge University Press:
- 19 June 2017, pp. 510-548
- Print publication:
- June 2017
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- Article
- Export citation
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We analyze the set-theoretic strength of determinacy for levels of the Borel hierarchy of the form
$\Sigma _{1 + \alpha + 3}^0 $, for α < ω1. Well-known results of H. Friedman and D.A. Martin have shown this determinacy to require α + 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 + 3}^0 $ determinacy, for 
$\alpha < \omega _1^{CK} $. This yields a characterization of the levels of L by or at which winning strategies in these games must be constructed. When α = 0, we have the following concise result: The least θ so that all winning strategies in 
${\rm{\Sigma }}_4^0 $ games belong to Lθ+1 is the least so that 
$L_\theta \models {\rm{``}}{\cal P}\left( \omega \right)$ exists, and all wellfounded trees are ranked”.
