We consider generalizations of a well-known class of spaces, called by S. Mrówka, N ∪R, where R is an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbersN . We denote these generalizations by ψ = ψ(κ,R) for κ ≥ ω. Mrówka proved the interesting theorem that there exists an R such that |βψ(ω,R) \ ψ(ω,R)| = 1. In other words there is a unique free z-ultrafilt...

Our motivating question was whether all traces on a U-ultrapower of a C*-algebra A, where U is a non-principal ultrafilter on N, are necessarily U-limits of traces on A. We show that this is false so long as A has infinitely many extremal traces, and even exhibit a 22 א0 size family of such traces on the ultrapower. For this to fail even when A has finitely many traces implies that A contains o...

The following pcf results are proved: 1. Assume that κ > א0 is a weakly compact cardinal. Let μ > 2κ be a singular cardinal of cofinality κ. Then for every regular λ < pp+Γ(κ)(μ) there is an increasing sequence ⟨λi | i < κ⟩ of regular cardinals converging to μ such that λ = tcf( ∏ i<κ λi, <Jbd κ ). 2. Let μ be a strong limit cardinal and θ a cardinal above μ. Suppose that at least one of them h...

Motivated by Tukey classification problems and building on work in [4], we develop a new hierarchy of topological Ramsey spaces Rα, α < ω1. These spaces form a natural hierarchy of complexity, R0 being the Ellentuck space [6], and for each α < ω1, Rα+1 coming immediately afterRα in complexity. Associated with each Rα is an ultrafilter Uα, which is Ramsey for Rα, and in particular, is a rapid p-...

I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to what will call Cut-for-Formulas Cut-for-Sets Theorem: for a set F and binary relation ⊢ on P ( ) $\mathcal {P}(F)$ , if finitary, monotonic, satisfies cut formulas, then it also sets. deduce CF/CS from Ultrafilter twice; each proof uses different order-theoretic variant of Tukey-Teichmüller...

Criteria are obtained for a filter F of subsets of a set I to be an intersection of finitely many ultrafilters, respectively, finitely many κ-complete ultrafilters for a given uncountable cardinal κ. From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the Loś-Eda theorem (characterizing h...

S OF PAPERS 899 Indecomposability is a weakening of the concept of measurability; indeed, inaccessible cardinals carrying indecomposable ultrafilters exhibit some of the strong reflection properties of measurable cardinals. Silver asked whether the two notions were actually equivalent for inaccessible cardinals, i.e., whether an inaccessible cardinal carrying an indecomposable ultrafilter must ...

Thanks also go to Professor Francesc Esteva, head of the Research Institute in Artificial Intelligence (IIIA) and to Professor`Angel Jorba, head of the Institute of Mathematics at the University of Barcelona (IMUB), for being so receptive to our needs and providing administrative support in different ways. The Oficina d'Activitats Institucionals of the University of Barcelona offered invaluable...

This paper contributes to the set-theoretic side of understanding Keisler’s order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality lcf(א0,D) of א0 modulo D, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as se...