The P-hierarchy of ultrafilters

We shall present the P-hierarchy of ultrafilters, that was posed by Andrzej Starosolski.
The P-hierarchy of ultrafilters is one of many ways to classify ultrafilters on natural numbers and it is composed of ℵ1 disjoint classes P(α) where α is ordinal number <ω1. The class P(1) is just a class of principal ultrafilters. The class P(2) is composed of P-points,  which were isolated by Rudin in order to prove non-homogeneity of the remainder of Cech-Stone compactification of natural numbers. Next, in higher classes of P-hierarchy, one can find ultrafilters with more and more complicated structures.

In this talk, we will disscuss relations between classes P(α) of P-hierarchy and other special types of ultrafilters, including: Baumgartner’s I-ultrafilters, thin ultrafilters, summable ultrafilters, and van der Waerden ultrafilters.