Related papers: Small Forcing Makes Any Cardinal Superdestructible
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,…
It is proved that the consistency strength of having definable tree property for successors of all regular cardinals is the consistency strength of having proper class many small large cardinals which are defined very similar to…
The paper settles the problem of the consistency of the existence of a single universal graph between a strong limit singular and its power. Assuming that in a model of $\mathbf{GCH}$ $\kappa$ is supercompact and the cardinals $\theta <…
A space X is kappa-resolvable (resp. almost kappa-resolvable) if it contains kappa dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X). Answering a problem raised by Juhasz, Soukup, and…
We consider compactness characterizations of large cardinals. Based on results of Benda \cite{b-sccomp}, we study compactness for omitting types in various logics. In $\bL_{\kappa, \kappa}$, this allows us to characterize any large cardinal…
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…
For every uncountable regular $\kappa$, we give two examples of proper posets which turn improper in some $\kappa$-closed forcing extension.
Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inaccessible in HOD.
Let omega be the first infinite ordinal (or the set of all natural numbers) with the usual order <. In section 1 we show that, assuming the consistency of a supercompact cardinal, there may exist an ultrapower of omega, whose cardinality is…
We present a version with non-definable forcing notions of Shelah's theory of iterated forcing along a template. Our main result, as an application, is that, if $\kappa$ is a measurable cardinal and $\theta<\kappa<\mu<\lambda$ are…
Building on work of Holy, L\"ucke and Njegomir \cite{MR3913154} on small embedding characterizations of large cardinals, we use some classical results of Baumgartner (see \cite{MR0384553} and \cite{MR0540770}), to give characterizations of…
A simple \(P_\lambda\)-point on a regular cardinal \(\kappa\) is a uniform ultrafilter on \(\kappa\) with a mod-bounded decreasing generating sequence of length \(\lambda\). We prove that if there is a simple $P_\lambda$-point ultrafilter…
A cardinal $\lambda$ satisfies a property P robustly if, whenever $\mathbb{Q}$ is a forcing poset and $|\mathbb{Q}|^+ < \lambda$, $\lambda$ satisfies P in $V^{\mathbb{Q}}$. We study the extent to which certain reflection properties of large…
We produce a model where every supercompact cardinal is $C^{(1)}$-supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in \cite{HMP} and provides a definitive answer to…
Assuming 0^sharp does not exist, kappa is an uncountable cardinal and for all cardinals lambda with kappa <= lambda < kappa^{+ omega}, 2^lambda = lambda^+, we present a ``mini-coding'' between kappa and kappa^{+ omega}. This allows us to…
We construct a model of the form $L[A,U]$ that exhibits the simplest structural behavior of $\sigma$-complete ultrafilters in a model of set theory with a single measurable cardinal $\kappa$ , yet satisfies $2^\kappa = \kappa^{++}$. This…
We establish the consistency of the failure of the diamond principle on a cardinal $\kappa$ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a…
Generalizing some earlier techniques due to the second author, we show that Menas' theorem which states that the least cardinal kappa which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not…
In this paper we are interested in finding and evaluating cardinal characteristics of the continuum that appear in large-scale topology, usually as the smallest weights of coarse structures that belong to certain classes (indiscrete,…
Given a Fra\"{i}ss\'{e} class $\mathcal{K}$ and an infinite cardinal $\kappa,$ we define a forcing notion which adds a structure of size $\kappa$ using elements of $\mathcal{K}$, which extends the Fra\"{i}ss\'{e} construction in the case…