Related papers: Consecutive singular cardinals and the continuum f…
We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible…
We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb…
We study the generalized dominating number $\mathfrak{d}_{\mu}$ at a singular cardinal $\mu$ of cofinality $\kappa$. We show two lower bounds: in ZFC, $\mathrm{cf}([\mu]^\kappa,\subseteq) \leq \mathfrak{d}_{\mu}$, and under mild…
We prove that successors of singular limits of strongly compact cardinals have the strong tree property. We also prove that aleph_{omega+1} can consistently satisfy the strong tree property.
The weakly compact reflection principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a weakly compact cardinal and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. The weakly compact…
We show a new proof for the fact that when $\kappa$ and $\lambda$ are infinite cardinals satisfying $\lambda ^ \kappa = \lambda$, the cofinality of the set of all functions from $\lambda$ to $\kappa$ ordered by everywhere domination is…
We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new ${<}\kappa$-sequences (for some regular $\kappa$). As an application, we show that consistently the following cardinal characteristics…
Assuming the consistency of ZFC with appropriate large cardinal axioms we produce a model of ZFC where $\aleph_\omega$ is a strong limit cardinal and the inner model $L(\mathcal{P}(\aleph_\omega))$ satisfies the following properties: (1)…
We provide a model where u(\kappa) < 2^{\kappa} for a supercompact cardinal \kappa. Garti and Shelah have provided a sketch of how to obtain such a model by modifying the construction in a paper of Dzamonja and Shelah; we provide here a…
We show that for many pairs of infinite cardinals $\kappa > \mu^+ > \mu$, $(\kappa^{+}, \kappa)\twoheadrightarrow (\mu^+, \mu)$ is consistent relative to the consistency of a supercompact cardinal. We also show that it is consistent,…
Assume $\kappa = \kappa^{< \kappa}$ (usually $\aleph_0$ or an inaccessible). We shall deal with iterated forcings preserving ${}^{\kappa>}{\rm Ord}$ and not collapsing cardinals along a linear order $L$. A sufficient condition for this,…
Recently the second author introduced combinatorial principles that characterize supercompactness for inaccessible cardinals but can also hold true for small cardinals. We prove that the proper forcing axiom PFA implies these principles…
We prove that superclub implies $\mathfrak{s}=\aleph_1$. More generally, superclub at a successor of a weakly compact cardinal implies $\mathfrak{s}_\kappa=\kappa^+$. Based on this statement, we separate tiltan from superclub at a successor…
Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We…
The current paper answers an open question of abs/1007.2426 We say that a countable model M characterizes an infinite cardinal kappa, if the Scott sentence of M has a model in cardinality kappa, but no models in cardinality kappa plus. If M…
In the first part of the paper, we show that if $\omega \le \kappa < \lambda$ are cardinals, $\kappa^{<\kappa} = \kappa$, and $\lambda$ is weakly compact, then in $V[\M(\kappa,\lambda)]$ the tree property at $\lambda =…
We show that, assuming GCH, if $\kappa$ is a Ramsey or a strongly Ramsey cardinal and $F$ is a class function on the regular cardinals having a closure point at $\kappa$ and obeying the constraints of Easton's theorem, namely,…
Assuming some large cardinals, a model of ZFC is obtained in which aleph_{omega+1} carries no Aronszajn trees. It is also shown that if lambda is a singular limit of strongly compact cardinals, then lambda^+ carries no Aronszajn trees.
Assuming $\kappa$ is a supercompact cardinal and $\lambda$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $\kappa=\aleph_\omega$ and $\lambda=\aleph_{\omega+1}.$
The paper is concerned with the existence of a universal graph at the successor of a strong limit singular mu of cofinality aleph_0. Starting from the assumption of the existence of a supercompact cardinal, a model is built in which for…