Crowded and selective ultrafilters under the Covering Property Axiom

by

Krzysztof Ciesielski, and J. Pawlikowski

J. Appl. Anal. 9(1) (2003), 19-55.

In the paper we formulate an axiom CPA_{prism}^{game}, 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 \omega1, while \continuum=\omega2: 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_{prism}^{game} implies the existence of crowded and selective ultrafilters as well as nonselective P-points. In addition we prove that under CPA_{prism}^{game} every selective ultrafilter is \omega1-generated. The paper is finished with the proof that CPA_{prism}^{game} holds in the iterated perfect set model.

It is known that the axiom CPA_{prism}^{game} captures the essence of the Sacks model concerning standard cardinal characteristics of continuum. This follows from a resent result of J. Zapletal who proved, assuming large cardinals, that for a nice'' cardinal invariant \kappa if \kappa<\continuum holds in any forcing extension than \kappa<\continuum follows already from CPA_{prism}^{game}.