PERFECT SUBSETS OF GENERALIZED BAIRE SPACES AND LONG GAMES. (9th January 2018)

• Record Type: Journal Article
• Authors: SCHLICHT, PHILIPP
• Is Part Of: Journal of symbolic logic. Volume 82:Number 4(2017)
• Page Start: 1317
• Page End: 1355
• Publication Date: 2018-01-09
• Subjects: 03E35, -- 03E60, -- 03E15, -- 03E15; generalized descriptive set theory, -- perfect set property, -- Banach–Mazur game, -- long games; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2017.44
• Languages: English
• ISSNs: 0022-4812

Abstract

Abstract: We extend Solovay's theorem about definable subsets of the Baire space to the generalized Baire space λ λ, where λ is an uncountable cardinal with λ <λ = λ . In the first main theorem, we show that the perfect set property for all subsets of λ λ that are definable from elements of λ Ord is consistent relative to the existence of an inaccessible cardinal above λ . In the second main theorem, we introduce a Banach–Mazur type game of length λ and show that the determinacy of this game, for all subsets of λ λ that are definable from elements of λ Ord as winning conditions, is consistent relative to the existence of an inaccessible cardinal above λ . We further obtain some related results about definable functions on λ λ and consequences of resurrection axioms for definable subsets of λ λ .