Related papers: Guessing models imply the singular cardinal hypoth…

Given a weakly compact cardinal $\kappa$, we give an axiomatization of intuitionistic first-order logic over $\mathcal{L}_{\kappa^+, \kappa}$ and prove it is sound and complete with respect to Kripke models. As a consequence we get the…

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 =…

The results of the previous version are impoved. This basically completes the study of consistency strength of various gaps between a strong limit singular cardinal of cofinality omega and its power under GCH type assumptions below.

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…

Assuming that there is no inner model with a Woodin cardinal, we obtain a characterization of $\lambda$-tall cardinals in extender models that are iterable. In particular we prove that in such extender models, a cardinal $\kappa$ is a tall…

We investigate the possibilities of global versions of Chang's Conjecture that involve singular cardinals. We show some $\mathrm{ZFC}$ limitations on such principles, and prove relative to large cardinals that Chang's Conjecture can…

Consider $(\kappa^{+++},\kappa^{++}) \twoheadrightarrow (\kappa^+,\kappa)$ where $\kappa$ is an uncountable regular cardinal. By a result of Shelah's we have $\operatorname{cof}(X \cap \kappa^{++}) = \kappa$ for almost all $X \subset…

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…

Local (first order) sentences, introduced by Ressayre, enjoy very nice decidability properties, following from some stretching theorems stating some remarkable links between the finite and the infinite model theory of these sentences. We…

Chang's Conjecture (CC) asserts that for every $F:[\omega_2]^{<\omega} \to \omega_2$, there exists an $X$ that is closed under $F$ such that $|X|=\omega_1$ and $|X \cap \omega_1| =\omega$. By classic results of Silver and Donder, CC is…

For infinite cardinals $\kappa,\lambda$ let $C(\kappa,\lambda)$ denote the class of all compact Hausdorff spaces of weight $\kappa$ and size $\lambda$. So $C(\kappa,\lambda)=\emptyset$ if $\kappa>\lambda$ or $\lambda>2^\kappa$. If F is a…

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,…

We present the classical theory of preservation of $\sqsubset$-unbounded families in generic extensions by ccc posets, where $\sqsubset$ is a definable relation of certain type on spaces of real numbers, typically associated with some…

Let $L$ be a countable language. We characterize, in terms of definable closure, those countable theories $\Sigma$ of $\mathcal{L}_{\omega_1, \omega}(L)$ for which there exists an $S_\infty$-invariant probability measure on the collection…

An $\omega_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $\omega_1$-compact space is $\sigma$-countably compact, i.e., the union of…

Inspired by work of Scheepers and Tall, we use properties defined by topological games to provide bounds for the cardinality of topological spaces. We obtain a partial answer to an old question of Bell, Ginsburg and Woods regarding the…

Shelah has provided sufficient conditions for an $L_{\omega_1, \omega}$-sentence $\psi$ to have arbitrarily large models and for a Morley-like theorem to hold of $\psi$. These conditions involve structural and set-theoretic assumptions on…

We develop a version of Cichon's diagram for cardinal invariants on the generalized Cantor space 2^kappa or the generalized Baire space kappa^kappa where kappa is an uncountable regular cardinal. For strongly inaccessible kappa, many of the…

The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…

We obtain results on the condensation principle called local club condensation. We prove that in extender models an equivalence between the failure of local club condensation and subcompact cardinals holds. This gives a characterization of…