Crowded and Selective Ultrafilters under the Covering Property Axiom

Abstract

In the paper we formulate an axiom CPA game prism , which is the most prominent version of the Covering Property Axiom CPA, and discuss several of its implications. In particular, we show that it implies that the following cardinal characteristics of continuum are equal to ω 1 , while c = ω 2 : the independence number i, the reaping number r, the almost disjoint number a, and the ultrafilter base number u. We will also show that CPA game prism implies the existence of crowded and selective ultrafilters as well as nonselective P -points. In addition we prove that under CPA game prism every selective ultrafilter is ω 1 -generated. The paper finishes with the proof that CPA game prism holds in the iterated perfect set model. * AMS classification numbers: Primary 54D80, 03E17; Secondary 03E35.

Uncountable γ-sets under axiom CPAcubegame

“…The Cantor set 2 ω will be denoted by C. We use the term Polish space for a complete separable metric space without isolated points. For a Polish space X, the symbol Perf(X) will denote the collection of all subsets of X homeomorphic to C. We will consider Perf(X) to be ordered by inclusion.Axiom CPA game cube was first formulated by Ciesielski and Pawlikowski in [4]. (See also [6].)…”

“…It is a simpler version of a Covering Property Axiom CPA which holds in the iterated perfect set model. (See [4] or [6].) In order to formulate CPA game cube we need the following terminology and notation.…”

The Covering Property Axiom, CPA

“…In this section, which in part is based on [34], we will show that CPA cube implies that every selective ultrafilter is generated by ω 1 sets and that the reaping number r is equal to ω 1 . The actual construction of a selective ultrafilter will require a stronger version of the axiom and will be done in Theorem 5.3.3.…”

“…11.5]) though it seems that the proof of this result was never provided. The argument presented below comes from [34]. …”

“…The next three sections are based on our paper [34]. We will use here the terminology introduced in Preliminaries Section.…”