Abstract

A p-point is a nonprincipal ultrafilter on the set N of natural numbers which has the property that for every countable family of filter sets there is a pseudointersection in the filter, i.e. a filter set which is almost contained in each set of the family. Equivalently, a p-point is an element of the Stone-Cech remainder beta(N) minus N whose neighborhood filter is closed under countable intersections.

It is well known that p-points “survive” various forcing iterations, that is: extending a universe V with certain forcing iterations P will result in a universe V’ in which all (or at least: certain well-chosen) p-points are still ultrafilter bases in the extension. This shows that the sentence “The continuum hypothesis is false, yet there are aleph1-generated ultrafilters, namely: certain p-points” is relatively consistent with ZFC .

In a joint paper with Diego Mejia and Saharon Shelah (still in progress) we construct ultrafilters on N which are, on the one hand, far away from being p-points (there is no Rudin-Keisler quotient which is a p-point), but on the other hand can survive certain forcing iterations adding reals but killing p-points. This shows that non-CH is consistent with small ultrafilter bases AND the nonexistence of p-points.