Let C denote any of the following cardinal characteristics of Boolean algebras: incomparability, spread, character, π-character, hereditary Lindelöf number, hereditary density. It is shown to be consistent that there exists a sequence 〈Bi : i < κ〉 of Boolean algebras and an ultrafilter D on κ such that

We consider the definability of mad families in vector spaces form ⨁ n < ω F where is a field cardinality ≤ ℵ 0 . show that there no analytic family subspaces when = 2 , partially answering question Smythe. Our proof relies on variant Mathias forcing restricted to certain idempotent ultrafilter whose existence follows from Glazer's Hindman's theorem.

Kirchberg asked in 2004 whether the commutant of L(H) in its (norm) ultrapower is trivial. Assuming the Continnuum Hypothesis, we prove that the answer depends on the choice of the ultrafilter. Let H be a separable infinite dimensional complex Hilbert space, fixed throughout. The purpose of this paper is to prove that, assuming the Continnuum Hypothesis, the commutant of L(H) (the algebra of bo...

We discuss conditions under which certain compactifications of topological spaces can be obtained by composing the ultrafilter space monad with suitable reflectors. In particular, we show that these inherit their categorical properties from monad. further observe various constructions such as prime open filter defined H. Simmons, closed compactification studied Bentley and Herrlich, well separa...

We develop a technology for investigation of natural forcing extensions of the model L(R) which satisfy such statements as “there is an ultrafilter” or “there is a total selector for the Vitali equivalence relation”. The technology reduces many questions about ZF implications between consequences of the axiom of choice to natural ZFC forcing problems.

In this paper we shall introduce the notions of point-closed, pointcompact, and m-saturated monotonic neighbourhood models. We will give some characterizations, and we will prove that the ultrafilter extension and the valuation extension of a model are m-saturated.

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic. Let (U) be the statement that a non-principal ultrafilter on N exists and let ACA0 be the higher order extension of ACA0. We show that ACA0 + (U) is Π2-conservative over ACA0 and thus that ACA0 +(U) is conservative over PA. Moreover, we provide a program extraction method and s...

We give a method for specifying ultrafilter equations and identify their projections on the set of profinite words. Let B be the set of languages captured by first-order sentences using unary predicates for each letter, arbitrary uniform unary numerical predicates and a predicate for the length of a word. We illustrate our methods by giving profinite equations characterizing B ∩Reg via ultrafil...

The notions of “labelled set” and “numerosity” are introduced to generalize the counting process of finite sets. The resulting numbers, called numerosities, are then used to develop nonstandard analysis. The existence of a numerosity function is equivalent to the existence of a selective ultrafilter, hence it is independent of the axioms of ZFC.