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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:
 300000 Other combinatorial set theory 3e+50 Continuum hypothesis and Martin’s axiom
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##### References:
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