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P-hierarchy on \(\beta \omega \). (English) Zbl 1156.03047

Summary: We classify ultrafilters on \(\omega \) with respect to sequential contours of different ranks. In this way we obtain an \(\omega_1\)-sequence \(\{\mathcal P_{\alpha }\}_{1\leq \alpha \leq \omega _1}\) of disjoint classes. We prove that non-emptiness of \(\mathcal P_{\alpha }\) for successor \(\alpha \geq 2\) is equivalent to the existence of a P-point. We investigate relations between the P-hierarchy and ordinal ultrafilters (introduced by J. E. Baumgartner), we prove that it is relatively consistent with ZFC that the successor classes (for \(\alpha \geq 2\)) of the P-hierarchy and ordinal ultrafilters intersect but are not the same.

MSC:

03E05 Other combinatorial set theory
03E35 Consistency and independence results
54F65 Topological characterizations of particular spaces
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