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]^{

Whether China should be more democratic has been a continuous debate among political theorists inside and outside of the country. Over the last decade or so, much scholarly discussion on the question surrounds the ideas of the political meritocrats, who hold that democratically elected officials should be constrained, if not replaced, by meritocratically selected officials, chosen by means of, for instance, examinations and peer recommendations. This institutional arrangement, most of them argue, is particularly suitable for the circumstances of China. The political meritocrats construct the dichotomy between democracy and political meritocracy by showing that political meritocracy is intimately connected to Chinese culture, and that democracy is in many ways Western. I argue, however, that this dichotomy is artificial and misleading. First, the ‘Chineseness’ of political meritocracy as well as ‘Westernness’ of democracy are often exaggerated by the political meritocrats. Second, the cultural considerations underpinning the Chineseness of meritocracy and Westernness of democracy are not taken as seriously by the political meritocrats’ view on the superiority of ideal meritocratic politics to non-ideal democratic politics, which renders the political meritocrats’ argument about the democracy–meritocracy dichotomy inconsistent.

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 {}^{

If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] which is the least ordinal with no presentation on [Formula: see text] computable from [Formula: see text]. If [Formula: see text] is a rigid Polish metric space and [Formula: see text] is any countable dense submetric space, then the Scott rank of [Formula: see text] is countable and in fact less than [Formula: see text].