There have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH, where α is a cardinal at most κ++. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = κ++, the maximum possible) and [1] (for α = κ+, after collapsing κ++). In addition, under stronger large cardinal hypotheses, one can handle the remaining...

The proof of the consistency of the existence of irregular ultra lters requires large cardinals. Foreman, Magidor, and Shelah forced the existence of a fully irregular ultra lter on a successor cardinal from a huge cardinal. That some large cardinal strength is necessary was shown by Ketonen (0) and Donder, Jensen, and Koppelberg (a measurable cardinal). We considerably improve this lower bound...

We consider maximality properties of inner models, elementary embeddings between them, and survey some connections through the concept of JJ onsson cardinal. In particular we give proofs of: Theorem Assume there is no inner model with a strong cardinal and K is the core model. If is a regular JJ onsson cardinal, then (i) + = +K ; (ii) f < j regular, + = +K g is stationary in ; (iii) 8A , A # ex...

Say that an elementary embedding j : N → M is cardinal preserving if CAR = CAR = CAR. We show that if PFA holds then there are no cardinal preserving elementary embeddings j : M → V . We also show that no ultrapower embedding j : V → M induced by a set extender is cardinal preserving, and present some results on the large cardinal strength of the assumption that there is a cardinal preserving j...

We construct models for the level by level equivalence between strong compactness and supercompactness containing failures of GCH at inaccessible cardinals. In one of these models, no cardinal is supercompact up to an inaccessible cardinal, and for every inaccessible cardinal δ, 2δ > δ++. In another of these models, no cardinal is supercompact up to an inaccessible cardinal, and the only inacce...

We determine the large cardinal consistency strength of the existence of a λ-supercompact cardinal κ such that GCH fails at λ. Indeed, we show that the existence of a λ-supercompact cardinal κ such that 2 ≥ θ is equiconsistent with the existence of a λ-supercompact cardinal that is also θ-tall. We also prove some basic facts about the large cardinal notion of tallness with closure.

Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of L(R) is absolute for proper forcing [Sch00]. Here, we study the indestructibility properties of remarkable cardinals. We show that if κ is remarkable, then there is a forcing extension in which the remarkability of κ becomes indestruct...

The approximate cardinal basis function (ACBF) preconditioning technique has been used to solve partial differential equations (PDEs) with radial basis functions (RBFs). In [31], a preconditioning scheme that is based upon constructing the least-squares approximate cardinal basis function from linear combinations of the RBF-PDE matrix elements has shown very attractive numerical results. This p...

The purpose of this paper is to introduce and investigate cardinal functions called pseudonet weight, weak net weight, and weak pseudonet weight. These are similar to but generally smaller than net weight. We look at how these cardinal functions relate to hereditary Lindelöf degree, hereditary density, and spread, and we study their behavior under products. An important and useful cardinal func...

We show that the consistency of the theory “ZF + DC + Every successor cardinal is regular + Every limit cardinal is singular + Every successor cardinal satisfies the tree property” follows from the consistency of a proper class of supercompact cardinals. This extends earlier results due to the author showing that the consistency of the theory “ZF + ¬ACω + Every successor cardinal is regular + E...