We classify ultrafilters on ω with respect to sequential contours (see [4]. [5]) of different ranks. In this way we obtain an ω1 sequence of disjoint classes. We prove that non-emptiness of for successor α ≥ 2 is equivalent to the existence of P-point. We investigate relations between P-hierarchy and ordinal ultrafilters (introduced by J. E. Baumgartner in [1]). we prove that it is relatively consistent with ZFC that the successor classes (for α ≥ 2) of P-hierarchy and ordinal ultrafilters intersect but are not the same.

An earlier paper [Starosolski A., P-hierarchy on βω, J. Symbolic Logic, 2008Logic, , 73(4), 1202Logic, -1214 investigated the relations between ordinal ultrafilters and the so-called P-hierarchy. The present paper focuses on the aspects of characterization of classes of ultrafilters of finite index, existence, generic existence and the Rudin-Keisler-order. MSC:2010 MSC: 03E20, 03E05

We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. Using that relation we prove (ZFC) that the class of strict J ω ω -ultrafilters, introduced by J. E. Baumgartner in Ultrafilters on ω, is empty. We translate the result to the language of < ∞ -sequences under an ultrafilter, investigated by C. Laflamme in A few special ordinal ultrafilters, to show that if there is an arbitrary long finite < ∞ -sequence under u than u is at least strict J ω ω+1 -ultrafilter.

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