A fairly quotable special, but still representative, case of our main result is that for 2 ≤ n < ω, there is a natural numberm(n) such that, the following holds. Assume GCH: If λ < μ are regular, there is a cofinality preserving forcing extension in which 2 = μ and, for all σ < λ ≤ κ < η such that η(+m(n)−1) ≤ μ, ((η)σ)→ ((κ)σ) (1)n η . This generalizes results of [3], Section 1, and the forcin...

We prove a revised version of Laver’s indestructibility theorem which slightly improves over the classical result. An application yields the consistency of (κ, κ) ։/ (א1,א0) when κ is supercompact. The actual proofs show that ω1-regressive Kurepatrees are consistent above a supercompact cardinal even though MM destroys them on all regular cardinals. This rather paradoxical fact contradicts the ...

Given a regular cardinal λ and λ many supercompact cardinals, we describe a type of forcing such that in the generic extension there is a cardinal κ with cofinality λ, the Singular Cardinal Hypothesis at κ fails, and the tree property holds at κ.

We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semiseparated scheme X. The approach generalizes and simplifies the method used by the author in [Gil04] and [Gil06] to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed...

We study several sorts of maximal almost disjoint families, both on a countable set and on uncountable, regular cardinals. We relate the associated cardinal invariants with bounding and dominating numbers and also with the uniformity of the meager ideal and some of its generalizations. 1. Who Are These Families? A Background Check Almost disjoint (ad) families have been a topic of interest in s...

in this paper we study the relationships existing between total measurability in variation and gould type fuzzy integrability (introduced and studied in [21]), giving a special interest on their behaviour on atoms and on finite unions of disjoint atoms. we also establish that any continuous real valued function defined on a compact metric space is totally measurable in the variation of a regula...

Assume ZFC $\mathsf {ZFC}$ . Let κ be a cardinal. A < ${\mathord {<}\hspace{1.111pt}\kappa }$ -ground is transitive proper class W modelling such that V generic extension of via forcing P ∈ $\mathbb {P}\in W$ cardinality The κ-mantle M $\mathcal {M}_\kappa$ the intersection all -grounds. We prove certain partial choice principles in are consequence being inaccessible/weakly compact, and some ot...

Let L[E] be an iterable tame extender model. We analyze to which extent L[E] knows fragments of its own iteration strategy. Specifically, we prove that inside L[E], for every cardinal κ which is not a limit of Woodin cardinals there is some cutpoint t < κ such that Jκ[E] is iterable above t with respect to iteration trees of length less than κ. As an application we show L[E] to be a model of th...

Uncountable superperfect forcing is tree forcing on regular uncountable cardinals κ with κ<κ = κ , using trees in which the heights of nodes that split along any branch in the tree form a club set, and such that any node in the tree with more than one immediate extension has measure-one-many extensions, where the measure is relative to some κ-complete, nonprincipal normal filter (or p-filter) F...

A compact set c in Rd is κ-round if for every point p ∈ ∂c there exists a closed ball that contains p, is contained in c, and has radius κdiam c. We show that, for any fixed κ > 0, the combinatorial complexity of the union of n κ-round, not necessarily convex objects in R3 (resp., in R4) of constant description complexity is O(n2+ε) (resp., O(n3+ε)) for any ε > 0, where the constant of proporti...