We introduce a hierarchy of large cardinals between weakly compact and measurable cardinals, that is closely related to the Ramsey-like cardinals introduced by Victoria Gitman in [Git11], and is based on certain infinite filter games, however also has a range of equivalent characterizations in terms of elementary embeddings. The aim of this paper is to locate the Ramsey-like cardinals studied b...

This paper investigates the relations K+ --t (a): and its variants for uncountable cardinals K. First of all, the extensive literature in this area is reviewed. Then, some possibilities afforded by large cardinal hypotheses are derived, for example, if K is measurable, then K + + (K + K + 1, a): for every a < K +. Finally, the limitations imposed on provability in ZFC by L and by relative consi...

We introduce the filter of locally closed unbounded sets and investigate its properties. As one application, we give a characterization of forcing notions that preserve stationary sets (in eo,). Also the local club filter and its variants lead to reflection principles for stationary sets. While the general reflection principle is related to Martin's Maximum of Foreman, Magidor and Shelah, the r...

We investigate the interaction between compactness principles and guessing in Radin forcing extensions. In particular, we show that any extension with respect to a measure sequence on [Formula: see text], if text] is weakly compact, then holds. This provides contrast well-known theorem of Woodin, who showed certain over suitably prepared ground model relative existence large cardinals, diamond ...

It is well known to generalize the meagre ideal replacing ℵ 0 by a (regular) cardinal λ > ℵ 0 and requiring the ideal to be λ +-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing ℵ 0 by λ, so requiring it to be (< λ)-complete. Of course, we would welcome additional properties generalizing the one...

Assume ZFC $\mathsf {ZFC}$ . Let κ be a cardinal. A < ${\mathord {<}\hspace{1.111pt}\kappa }$ -ground is transitive proper class W modelling such that V generic extension of via forcing P ∈ $\mathbb {P}\in W$ cardinality The κ-mantle M $\mathcal {M}_\kappa$ the intersection all -grounds. We prove certain partial choice principles in are consequence being inaccessible/weakly compact, and some ot...

in this note we characterize the compact weighted frobenius-perron operator $p$ on $l^1(sigma)$ and determine their spectra. we also show that every weakly compact weighted frobenius-perron operator on $l^1(sigma)$ is compact.

The following pcf results are proved: 1. Assume that κ > א0 is a weakly compact cardinal. Let μ > 2κ be a singular cardinal of cofinality κ. Then for every regular λ < pp+Γ(κ)(μ) there is an increasing sequence ⟨λi | i < κ⟩ of regular cardinals converging to μ such that λ = tcf( ∏ i<κ λi, <Jbd κ ). 2. Let μ be a strong limit cardinal and θ a cardinal above μ. Suppose that at least one of them h...

{V : V ∈ Vn} = X . Clearly, every Menger space is almost Menger and every almost Menger space is weakly Menger, but the converses do not hold (see Examples 2.1 and 2.2). On the study of weakly Menger spaces, almost Menger spaces and Menger spaces, the readers can see the references [2, 3, 4, 5, 6]. Here we investigate the relationships among almost Menger spaces, weakly Menger spaces and Menger...