𝑃-points with countably many constellations

Rosen, Ned I.

doi:10.1090/S0002-9947-1985-0792813-8

$P$-points with countably many constellations
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Trans. Amer. Math. Soc. 290 (1985), 585-596 Request permission

Abstract:

If the continuum hypothesis $({\text {CH}})$ is true, then for any $P$ point ultrafilter $D$ (on the set of natural numbers) there exist initial segments of the Rudin-Keisler ordering, restricted to (isomorphism classes of) $P$ points which lie above $D$, of order type ${\aleph _1}$. In particular, if $D$ is an ${\text {RK}}$-minimal ultrafilter, then we have $({\text {CH}})$ that there exist $P$-points with countably many constellations.

References

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Andreas Blass, Ultrafilter mappings and their Dedekind cuts, Trans. Amer. Math. Soc. 188 (1974), 327–340. MR 351822, DOI 10.1090/S0002-9947-1974-0351822-6
Andreas Blass, Some initial segments of the Rudin-Keisler ordering, J. Symbolic Logic 46 (1981), no. 1, 147–157. MR 604888, DOI 10.2307/2273266

Model theory

Über Ultrafilter und Nichtstandardmodelle mit vorgeschriebener Verteilung der Konstellationen

Christian Puritz, Ultrafilters and standard functions in non-standard arithmetic, Proc. London Math. Soc. (3) 22 (1971), 705–733. MR 289283, DOI 10.1112/plms/s3-22.4.705
Ned I. Rosen, Weakly Ramsey $P$ points, Trans. Amer. Math. Soc. 269 (1982), no. 2, 415–427. MR 637699, DOI 10.1090/S0002-9947-1982-0637699-4