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Chains of P-points

Part of:Set theoryFairly general properties

Published online by Cambridge University Press:  18 February 2019

Dilip Raghavan and
Jonathan L. Verner
Dilip Raghavan
Affiliation:
Department of Mathematics, National University of Singapore, Singapore 119076 Email: dilip.raghavan@protonmail.com
Jonathan L. Verner
Affiliation:
Department of Logic, Faculty of Arts, Charles University, nám. Jana Palacha 2, 116 38 Praha 1 Email: jonathan.verner@ff.cuni.cz
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Abstract

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It is proved that the Continuum Hypothesis implies that any sequence of rapid P-points of length ${<}\mathfrak{c}^{+}$ that is increasing with respect to the Rudin–Keisler ordering is bounded above by a rapid P-point. This is an improvement of a result from B. Kuzeljevic and D. Raghavan. It is also proved that Jensen’s diamond principle implies the existence of an unbounded strictly increasing sequence of P-points of length $\unicode[STIX]{x1D714}_{1}$ in the Rudin–Keisler ordering. This shows that restricting to the class of rapid P-points is essential for the first result.

Keywords

Rudin–Keisler orderultrafilterP-point

MSC classification

Primary:03E50: Continuum hypothesis and Martin's axiom
Secondary:03E05: Other combinatorial set theory54D80: Special constructions of spaces (spaces of ultrafilters, etc.)

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 4 , December 2019, pp. 856 - 868
Copyright
© Canadian Mathematical Society 2019 

Footnotes

Author D. R. was partially supported by National University of Singapore research grant number R-146-000-211-112. Author J. L. V. was supported by the joint FWF-GAČR grant no. 17-33849L, by the Progres grant Q14, and by grant number R-146-000-211-112 to author D. R. from the National University of Singapore.

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