Related papers: Jonsson Cardinals, Erd\H{o}s Cardinals, and the Co…
Assume $\boldsymbol{\Delta}^1_3$-determinacy. It is shown that for any $x \geq_T M_1^{\#}$, $\mathrm{HOD}^{L[x]}$ is a model of GCH, and in fact, it is a Jensen-Steel core model up to $\omega_2^{L[x]}$.
After small forcing, almost every strongness embedding is the lift of a strongness embedding in the ground model. Consequently, small forcing creates neither strong nor Woodin 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.
If cf(kappa) = kappa, kappa^+< cf(lambda) = \lambda, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing…
In [Bon20], model theoretic characterizations of several established large cardinal notions were given. We continue this work, by establishing such characterizations for Woodin cardinals (and variants), various virtual large cardinals, and…
We prove in the theory "ZFC + there is no inner model with a Woodin cardinal" that there is a universal weasel. This shows in particular that one doesn't have to assume the existence of large cardinals in V to prove the existence of some…
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,…
Carsten Thomassen in 1989 conjectured that if a graph has minimum degree more than the number of atoms in the universe ($\delta(G)\ge 10^{10^{10}}$), then it contains a pillar, which is a graph that consists of two vertex-disjoint cycles of…
Let GCH hold and let $j:V\longrightarrow M$ be a definable elementary embedding such that $crit(j)=\kappa$, $^{\kappa}M\subseteq M$ and $\kappa^{++}=\kappa_{M}^{++}$. H. Woodin proved that there is a cofinality preserving generic extension…
The pinning down number $ {pd}(X)$ of a topological space $X$ is the smallest cardinal $\kappa$ such that for any neighborhood assignment $U:X\to \tau_X$ there is a set $A\in [X]^\kappa$ with $A\cap U(x)\ne\emptyset$ for all $x\in X$.…
Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan $\ast$-derivations, and to prove some results involving these mappings.…
We define a generic Vop\v{e}nka cardinal to be an inaccessible cardinal $\kappa$ such that for every first-order language $\mathcal{L}$ of cardinality less than $\kappa$ and every set $\mathscr{B}$ of $\mathcal{L}$-structures, if…
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, and is based on certain infinite filter games, however also has…
We characterize several large cardinal notions by model-theoretic properties of extensions of first-order logic. We show that $\Pi_n$-strong cardinals, and, as a corollary, ``Ord is Woodin" and weak Vop\v{e}nka's Principle, are…
We survey some recent results on the validity of Jensen's diamond principle at successor cardinals. We also discuss weakening of this principle such as club guessing, and anti-diamond principles such as uniformization. A collection of open…
We show that if the existence of a supercompact cardinal $\kappa$ with a weakly compact cardinal $\lambda$ above $\kappa$ is consistent, then the following are consistent as well (where $\mathfrak{t}(\kappa)$ and $\mathfrak{u}(\kappa)$ are…
A 27 years old and still open problem of Juhasz and van Mill asks whether there exists a cardinal kappa such that every regular dense in itself countably compact space has a dense in itself subset of cardinality at most kappa. We give a…
We introduce (super-$C^{(\infty)}$-)Laver-generic large cardinal axioms for extendibility ((super-$C^{(\infty)}$-)LgLCAs for extendible, for short), and show that most of the previously known consequences of the…
A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure has a…
In this paper we prove the equiconsistency of ``Every omega_1 tree which is first order definable over H_{omega_1} has a cofinal branch'' with the existence of a Pi^1_1 reflecting cardinal. The proof uses a definable version of Ramsey…