Σ1(κ)-DEFINABLE SUBSETS OF H(κ+). (8th September 2017)

• Record Type: Journal Article
• Authors: LÜCKE, PHILIPP; SCHINDLER, RALF; SCHLICHT, PHILIPP
• Is Part Of: Journal of symbolic logic. Volume 82:Number 3(2017)
• Page Start: 1106
• Page End: 1131
• Publication Date: 2017-09-08
• Subjects: 03E45, -- 03E47, -- 03E55; Σ1-definability, -- lightface formulas, -- large cardinals, -- well-orders, -- club filter, -- nonstationary ideal, -- Bernstein sets, -- iterated ultrapowers, -- iterated generic ultrapowers; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2017.36
• Languages: English
• ISSNs: 0022-4812

Abstract

Abstract: We study Σ1 ( ω 1 )-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1 -formula with parameter ω 1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1 ( ω 1 )-definable, the set of all stationary subsets of ω 1 is not Σ1 ( ω 1 )-definable and the complement of every Σ1 ( ω 1 )-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1 ( ω 1 )-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1 ( ω 1 )-definable well-ordering of H( ω 2 ) and the existence of a Δ1 ( ω 1 )-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ . We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1 ( ω 1 )-definable uniformization of the club filter on ω 1 . Moreover, we prove a perfect set theorem for Σ1 ( ω 1 )-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω 1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin's ℙmax -forcing. Finally, we also prove variants of some of these results for Σ1 ( κ )-definable subsets of κ κ, in the case where κ itself has certain large cardinal properties.