How high can Baumgartner’s $${\mathcal{I}}$$ I -ultrafilters lie in the P-hierarchy?

Abstract

Under the continuum hypothesis we prove that for any tall P-ideal I on ω and for any ordinal γ ≤ ω 1 there is an I-ultrafilter in the sense of Baumgartner, which belongs to the class P γ of the P-hierarchy of ultrafilters. Since the class of P 2 ultrafilters coincides with the class of P-points, our result generalizes the theorem of Flašková, which states that there are I-ultrafilters which are not P-points.

Continuous extension of maps between sequential cascades

“…Specifically, an ultrafilter u ∈ P α if for each β < α there exists a monotone sequential contour C of rank β, such that C ⊂ u and there is no monotone sequential contour of rank α contained in u. It appears that the class P 2 is precisely that of P -points ( [17], [19], [16]). The continuous extension (Theorem 7.2) implies that if u ∈ P α then there is γ ≤ α such that f (u) ∈ P γ .…”