相关论文: A small ultrafilter number at smaller cardinals
We obtain a small ultrafilter number at $\aleph_{\omega_1}$. Moreover, we develop a version of the overlapping strong extender forcing with collapses which can keep the top cardinal $\kappa$ inaccessible. We apply this forcing to construct…
In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which $\aleph_{\omega+1}$ carries a uniform ultrafilter that is $\theta$-indecomposable for every uncountable cardinal $\theta<\aleph_\omega$. In this…
We prove the consistency of a singular cardinal $\lambda$ with small value of the ultrafilter number $u_\lambda$, and arbitrarily large value of $2^\lambda$.
Starting from a supercompact cardinal we build a model in which $2^{\aleph_{\omega_1}}=2^{\aleph_{\omega_1+1}}=\aleph_{\omega_1+3}$ but there is a jointly universal family of size $\aleph_{\omega_1+2}$ of graphs on $\aleph_{\omega_1+1}$.…
Assuming an abstract comparison principle called the Ultrapower Axiom, which is motivated by the comparison process of inner model theory and generalizes the statement that the Mitchell order is linear on normal ultrafilters, we…
In this paper, we characterize the possible cofinalities of the least $\lambda$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $\delta$, that carries a $\lambda$-complete uniform ultrafilter, it is…
We continue investigations of reasonable ultrafilters on uncountable cardinals defined in Shelah math.LO/0407498 and studied also in math.LO/0605067. We introduce a general scheme of generating a filter on lambda from filters on smaller…
A ccc-generically supercompact cardinal $\kappa$ can be smaller than or equal to the continuum. On the other hand, such a cardinal $\kappa$ still satisfies diverse largeness properties, like that it is a stationary limit of ccc-generically…
We prove that consistently there is a singular cardinal $\kappa$ of uncountable cofinality such that $2^\kappa$ is weakly inaccessible, and every regular cardinal strictly between $\kappa$ and $2^\kappa$ is the character of some uniform…
This note shows how recent work of Eskew and Hayut can be combined with a method of Raghavan and Shelah to show the consistency of $\mathfrak u_\kappa<2^\kappa$ for $\aleph_3\leq \kappa<\aleph_{\omega}$, from a huge cardinal.
We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V is a model…
It is shown that the consistency strength of ZF + DC + "the closed unbounded ultrafilter on omega_1 is an ultrafilter" is exactly ZFC + one measurable cardinal.
We develop the theory of cofinal types of ultrafilters over measurable cardinals and establish its connections to Galvin's property. We generalize fundamental results from the countable to the uncountable, but often in surprisingly…
We continue investigations of reasonable ultrafilters on uncountable cardinals defined in math.LO/0407498. We introduce stronger properties of ultrafilters and we show that those properties may be handled in lambda-support iterations of…
All ultrafilters under consideration here are non-principal ultrafilters on the set omega of natural numbers. We are concerned with the possible cofinalities of ultrapowers of omega with respect to such ultrafilters. We show that no…
We study ultrafilters on regular uncountable cardinals, with a primary focus on $\omega_1$, and particularly in relation to the Tukey order on directed sets. Results include the independence from ZFC of the assertion that every uniform…
Answering a question of Ketonen from the late 1970's, it is proved that a weakly compact cardinal carrying an indecomposable ultrafilter need not be measurable. The result is obtained by analyzing the limit of a decreasing sequence of…
We are interested in generalizing part of the theory of ultrafilters on omega to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal.
We characterize ultrafilter convergence and ultrafilter compactness in linearly ordered and generalized ordered topological spaces. In such spaces, and for every ultrafilter $D$, the notions of $D$-compactness and of $D$-pseudocompactness…
This short note contains the proofs of two small but somewhat surprising results about ultrafilters on $\mathbb{N}$: 1. strongly summable ultrafilters are rapid, 2. every rapid ultrafilter induces a closed left ideal of rapid ultrafilters.…