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How high can Baumgartner’s \(\mathcal{I}\)-ultrafilters lie in the P-hierarchy? (English) Zbl 1373.03074
It is shown that if the continuum hypothesis is assumed, then for any tall P-ideal \(I\) on \(\omega\) containing all singletons, and any ordinal \(\gamma\) with \(2 \leq \gamma \leq \omega_1\), there are two Rudin-Keisler incomparable ultrafilters \(U_1\) and \(U_2\) on \(\omega\) such that for \(j = 1, 2\), (a) \(U_j\) is an \(I\)-ultrafilter (meaning that for each function \(f : \omega \rightarrow \omega\), there is \(A \in U_j\) with \(f [A]\) in \(I\)), and (b) \(U_j\) belongs to the class \({\mathcal P}_\gamma\) in the P-hierarchy of ultrafilters.

MSC:
03E05 Other combinatorial set theory
03E50 Continuum hypothesis and Martin’s axiom
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