Let κ be a singular cardinal of countable cofinality, 〈κn : n < ω〉 a sequence of regular cardinals which is increasing and cofinal in κ. Using a scale, we define a mapping μ from ∏ n P(κn) to P(κ+) which relates tight stationarity on κ to the usual notion of stationarity on κ+. We produce a model where all subsets of κ+ are in the range of μ for some κ a singular. Using a version of the diagona...

The optical properties of a number of supercompact ultraviolet luminous galaxies (UVLG), recently discovered in the local Universe matching GALEX and Sloan Digital Sky Survey (SDSS) data, are discussed. Detailed re-analysis of the SDSS data for these and other similar but nearer galaxies shows that their surface brightness radial profile in both R and u bands is in most cases well described by ...

The motivation for this paper is the following: In [Fuc08] I showed that it is inconsistent with ZFC that the maximality principle for closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the maximality principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regula...

We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional “L-like” combinatorial principles. In particular, this model satisfies the following properties: 1. ♦δ holds for every successor and Mahlo cardinal δ. 2. There is a stationary subset S of the least supercompact cardinal κ0 such that for e...

We provide a lower bound for the consistency strength of the hypothesis proposed by S. Kamo: ∃κ > א0 s(κ) ≥ κ. 0. Introduction. In [3] ,[10] cardinal invariants on ω are generalized for uncountable regular cardinals and it is shown that some of their properties still hold true for their uncountable counterparts.However,the role of the splitting number s(κ) (for the definition see below) changes...

A space X is said to be κ-resolvable (resp. almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable iff it is ∆(X)-resolvable, where ∆(X) = min{|G| : G 6= ∅ open}. We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = mi...

The relations M(κ, λ, µ) → B [resp. B(σ)] meaning that if A ⊂ [κ] λ with |A| = κ is µ-almost disjoint then A has property B [resp. has a σ-transversal] had been introduced and studied under GCH in [EH]. Our two main results here say the following: Assume GCH and ̺ be any regular cardinal with a supercompact [resp. 2-huge] cardinal above ̺. Then there is a ̺-closed forcing P such that, in V P , we ...

The normal Moore space conjecture asserts that normal Moore spaces are metrizable. Nyikos has proven the consistency (from the existence of a strongly compact cardinal) of the conjecture holding and Fleissner has proven that at least a measurable cardinal is needed to prove the consistency. Although extremely elegant, Nyikos’ proof relies on Kunen’s proof of the consistency of the product measu...

We give an exposition in modern language (and using partial orders) of Jech’s method for obtaining models where successor cardinals have large cardinal properties. In such models, the axiom of choice must necessarily fail. In particular, we show how, given any regular cardinal and a large cardinal of the requisite type above it, there is a symmetric extension of the universe in which the axiom ...

One of the most important goals of set theorists over the last few years has been to re-prove old results which previously had used very strong assumptions from hypotheses which, at least prima facie, are weaker. Examples of these abound, including, but certainly not limited to, the work of Woodin and Cummings (see [3]) on the Singular Cardinals Problem, in which results previously obtained by ...