More definable combinatorics around the first and second uncountable cardinals

Journal of Mathematical Logic 23 (3) (2023)
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Abstract

Assume [Formula: see text]. If [Formula: see text] is an ordinal and X is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every [Formula: see text] and function [Formula: see text], there is a club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. For every [Formula: see text] and function [Formula: see text], there is an [Formula: see text]-club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of [Formula: see text]. [Formula: see text]. [Formula: see text]. [Formula: see text]. It will also be shown that [Formula: see text] has the Jónsson property: For every [Formula: see text], there is an [Formula: see text] with [Formula: see text] so that [Formula: see text].

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William Chan
University of Manchester

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References found in this work

Ordinal definability and combinatorics of equivalence relations.William Chan - 2019 - Journal of Mathematical Logic 19 (2):1950009.
A dichotomy for the definable universe.Greg Hjorth - 1995 - Journal of Symbolic Logic 60 (4):1199-1207.
The cardinals below |[ω1]<ω1|.W. Hugh Woodin - 2006 - Annals of Pure and Applied Logic 140 (1-3):161-232.

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