Now it is obvious from the usual definitions that an inaccessible κ is subtle iff every κ-tree T satisfies STP(T ), for which we shall just write κ-STP, iff the complete binary tree 2<κ satisfies STP(2<κ), and similarly for ineffability (and one can take this as a definition if unfamiliar with the concepts). By [Mit73] one can collapse a weakly compact (a Mahlo) cardinal onto ω2 such that in th...

We construct models containing exactly one supercompact cardinal in which level by level inequivalence between strong compactness and supercompactness holds. In each model, above the supercompact cardinal, there are finitely many strongly compact cardinals, and the strongly compact and measurable cardinals precisely coincide. Say that a model containing supercompact cardinals satisfies level by...

We introduce and study a new topology on trees, that we call the countably coarse wedge topology. Such is strictly finer than it turns every chain complete, rooted tree into Fréchet–Urysohn, compact topological space. show rôle of such in theory weakly Corson Valdivia compacta. In particular, give first example space T whose closed subspace Valdivia, yet not Corson. This answers question due to...

We construct three models in which there are different relationships among the classes of strongly compact, strong, and non-strong tall cardinals. In the first two of these models, the strongly compact and strong cardinals coincide precisely, and every strongly compact/strong cardinal is a limit of non-strong tall cardinals. In the remaining model, the strongly compact cardinals are precisely c...

The Levy-Solovay Theorem [LevSol67] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong card...

After forcing which admits a very low gap—and this includes many of the forcing iterations, such as the Laver preparation, which are commonly found in the large cardinal context—every embedding j : V [G] → M [j(G)] in the extension which satisfies a mild closure condition is the lift of an embedding j : V → M in the ground model. In particular, every ultrapower embedding in the extension lifts ...