相关论文: Cardinal characteristics at {\kappa} in a small u(…

We prove that if there are $\mathfrak c$ incomparable selective ultrafilters then, for every infinite cardinal $\kappa$ such that $\kappa^\omega=\kappa$, there exists a group topology on the free Abelian group of cardinality $\kappa$…

For an inaccessible cardinal $\kappa$, the super tree property (ITP) at $\kappa$ holds if and only if $\kappa$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor…

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals…

Let kappa be an uncountable regular cardinal. Assuming 2^kappa=kappa^+, we show that the clone lattice on a set of size kappa is not dually atomic.

We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $\omega$ of asymptotic density $0$. We obtain…

A stationary subset $S$ of a regular uncountable cardinal $\kappa$ {\it reflects fully} at regular cardinals if for every stationary set $T \subseteq \kappa$ of higher order consisting of regular cardinals there exists an $\alpha \in T$…

For $\kappa$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^\kappa\kappa$ and ${}^\kappa2$ (endowed with the ${<}\kappa$-box topology) have been relatively well-studied, but less is known about the case where…

We prove a revised version of Laver's indestructibility theorem which slightly improves over the classical result. An application yields the consistency of $(\kappa^+,\kappa)\notcc(\aleph\_1,\aleph\_0)$ when $\kappa$ is supercompact. The…

The purpose of this paper is to provide an introductory overview of the large cardinal hierarchy in set theory. By a large cardinal, we mean any cardinal $\kappa$ whose existence is strong enough of an assumption to prove the consistency of…

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 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 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…

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $\kappa \ge \omega_2$, $\textsf{ISP}(\kappa)$ implies that $\textsf{SCH}$ holds above $\kappa$, and (3) forcing posets…

We prove that the existence of a complete metric space of cardinality at most $2^{\kappa}$ admitting Kuratowski partition is a consequence of $\kappa$ being the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$.

We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, we show that if $\lambda \geq 2^\kappa$, then (a) $I_{\kappa, \lambda}$ is…

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 show that some of the most prominent large cardinal notions can be characterized through the validity of certain combinatorial principles at $\omega_2$ in forcing extensions by the pure side condition forcing introduced by Neeman. The…

We investigate the circumstances under which there exist a singular cardinal $\mu$ and a short $(\kappa, \mu)$-extender $E$ witnessing "$\kappa$ is $\mu$-strong", such that $\mu$ is singular in $\Ult(V, E)$.

For an uncountable regular cardinal \kappa we let \nabla_\kappa(A) be the statement that A \subset \kappa and for all regular \theta > \kappa, the set of all X \in [\theta]^<\kappa such that X \cap \kappa \in \kappa and otp(X \cap OR) is a…

Jech proved that every partially ordered set can be embedded into the cardinals of some model of $ZF$. We extend this result to show that every partially ordered set can be embedded into the cardinals of some model of $ZF+DC_{<\kappa}$ for…