$G_{\delta \sigma }$ GAMES AND INDUCTION ON REALS. (13th December 2021)

• Record Type: Journal Article
• Authors: AGUILERA, J. P.; WELCH, P. D.
• Is Part Of: Journal of symbolic logic. Volume 86:Number 4(2021)
• Page Start: 1676
• Page End: 1690
• Publication Date: 2021-12-13
• Subjects: 03B30 -- 03D60 -- 03D70 -- 03E60 -- 91A44; determinacy -- games on reals -- inductive definitions; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2021.66
• Languages: English
• ISSNs: 0022-4812

Abstract

Abstract: It is shown that the determinacy of $G_{\delta \sigma }$ games of length $\omega ^2$ is equivalent to the existence of a transitive model of ${\mathsf {KP}} + {\mathsf {AD}} + \Pi _1\textrm {-MI}_{\mathbb {R}}$ containing $\mathbb {R}$ . Here, $\Pi _1\textrm {-MI}_{\mathbb {R}}$ is the axiom asserting that every monotone $\Pi _1$ operator on the real numbers has an inductive fixpoint.