Decomposable Ultrafilters and Possible Cofinalities

Decomposable Ultrafilters and Possible Cofinalities

We use Shelah’s theory of possible cofinalities in order to solve some problems about ultrafilters. Theorem 1. Suppose that λ is a singular cardinal, λ′ < λ, and the ultrafilter D is κ-decomposable for all regular cardinals κ with λ′ < κ < λ. Then D is either λ-decomposable, or λ-decomposable. Corollary 2. If λ is a singular cardinal, then an ultrafilter is (λ, λ)-regular if and only if it is e...

A Connection between Decomposability of Ultrafilters and Possible Cofinalities

We introduce the decomposability spectrum KD = {λ ≥ ω|D is λ-decomposable} of an ultrafilter D, and show that Shelah’s pcf theory influences the possible values KD can take. For example, we show that if a is a set of regular cardinals, μ ∈ pcf a, the ultrafilter D is |a|-complete and KD ⊆ a, then μ ∈ KD. As a consequence, we show that if λ is singular and for some λ < λ KD contains all regular ...

Dropping cofinalities

Our aim is to present constructions in which some of the cofinalities drop down, i.e. the generators of PCF structure are far a part. 1 Some Preliminary Settings Let λ0 < κ0 < λ1 < κ1 < ... < λn < κn < ...., n < ω be a sequence of cardinals such that for each n < ω • λn is λ +n+2 n +2 n strong as witnessed by an extender Eλn • κn is κ +n+2 n +2 n strong as witnessed by an extender Eκn Let κ = ⋃...

Dropping cofinalities and gaps

Our aim is to show the consistency of 2κ ≥ κ+θ, for any θ < κ starting with a singular cardinal κ of cofinality ω so that ∀n < ω∃α < κ(o(α) = α). Dropping cofinalities technics are used for this purpose. 1 א1-gap and infinite drops in cofinalities Let κ be a singular cardinal of cofinality ω such that for each γ < κ and n < ω there is α, γ < α < κ, such that o(α) = α. We fix a sequence of cardi...

Cofinalities of Linear Orders