William Chan

Within the determinacy setting, ${\mathscr {P}({\omega _1})}$ is regular (in the sense of cofinality) with respect to many known cardinalities and thus there is substantial evidence to support the conjecture that ${\mathscr {P}({\omega _1})}$ has globally regular cardinality. However, there is no known information about the regularity of ${\mathscr {P}(\omega _2)}$. It is not known if ${\mathscr {P}(\omega _2)}$ is even $2$ -regular under any determinacy assumptions. The article will provide the following evidence that ${\mathscr {P}(\omega _2)}$ may possibly be ${\omega _1}$ -regular: Assume $\mathsf {AD}^+$. If $\langle A_\alpha : \alpha < {\omega _1} \rangle $ is such that ${\mathscr {P}(\omega _2)} = \bigcup _{\alpha < {\omega _1}} A_\alpha $, then there is an $\alpha < {\omega _1}$ so that $\neg (|A_\alpha | \leq |[\omega _2]^{

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].

Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $, $\vee $, $\forall ^{\mathbb {R}}$, continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $. Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $. Then the countable length everywhere club uniformization holds for $\kappa $ : For every relation $R \subseteq {}^{