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Chains of P-points

Part of: Fairly general properties Set theory

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: [email protected]
Jonathan L. Verner
Affiliation:
Department of Logic, Faculty of Arts, Charles University, nám. Jana Palacha 2, 116 38 Praha 1 Email: [email protected]
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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 theory 54D80: Special constructions of spaces (spaces of ultrafilters, etc.)
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.

References

Blass, A., Orderings of ultrafilters. Ph.D. thesis, Harvard University, 1970.Google Scholar
Blass, A., The Rudin–Keisler ordering of P-points . Trans. Amer. Math. Soc. 179(1973), 145–166. https://doi.org/10.2307/1996495 Google Scholar
Dobrinen, N. and Todorcevic, S., Tukey types of ultrafilters . Illinois J. Math. 55(2013), no. 3, 907–951.Google Scholar
Frolík, Z., Sums of ultrafilters . Bull. Amer. Math. Soc. 73(1967), 87–91. https://doi.org/10.1090/S0002-9904-1967-11653-7 Google Scholar
Isbell, J. R., The category of cofinal types. II . Trans. Amer. Math. Soc. 116(1965), 394–416. https://doi.org/10.2307/1994124 Google Scholar
Katětov, M., Products of filters . Commentationes Mathematicae Universitatis Carolinae 009(1968), no. 1, 173–189.Google Scholar
Kuzeljevic, B. and Raghavan, D., A long chain of P-points . J. Math. Log. 18(2018), no. 1, 1850004, 38 pp. https://doi.org/10.1142/S0219061318500046 Google Scholar
Laflamme, C., Forcing with filters and complete combinatorics . Ann. Pure Appl. Logic 42(1989), no. 2, 125–163. https://doi.org/10.1016/0168-0072(89)90052-3 Google Scholar
Laflamme, C. and Zhu, J.-P., The Rudin-Blass Ordering of Ultrafilters . J. Symbolic Logic 63(1998), no. 2, 584–592. https://projecteuclid.org:443/euclid.jsl/1183745522 Google Scholar
Milovich, D., Tukey classes of ultrafilters on omega . Topology Proceedings 32(2008), 351–362. arxiv:0807.3978v1 Google Scholar
Mokobodzki, G., Ultrafiltres rapides sur ℕ. Construction d’une densité relative de deux potentiels comparables . In: Séminaire de Théorie du potentiel . Secrétariat mathématique, Paris, 1969, pp. 1–22.Google Scholar
Raghavan, D. and Shelah, S., On embedding certain partial orders into the P-points under Rudin–Keisler and Tukey reducibility . Trans. Amer. Math. Soc. 369(2017), no. 6, 4433–4455. https://doi.org/10.1090/tran/6943 Google Scholar
Raghavan, D. and Todorcevic, S., Cofinal types of ultrafilters . Ann. Pure Appl. Logic 163(2012), no. 3, 185–199. https://doi.org/10.1016/j.apal.2011.08.002 Google Scholar
Rosen, N. I., P-points with countably many constellations . Trans. Amer. Math. Soc. 290(1985), no. 2, 585–596. https://doi.org/10.2307/2000300 Google Scholar
Rudin, M. E., Types of ultrafilters . In: Topology seminar . Ann. of Math. Studies, 60, Princeton University Press, Princeton, NJ, 1966, pp. 147–151.Google Scholar
Rudin, M. E., Partial orders on the types in 𝛽N . Trans. Amer. Math. Soc. 155(1971), 353–362. https://doi.org/10.2307/1995690 Google Scholar
Rudin, W., Homogeneity problems in the theory of Čech compactifications . Duke Math. J. 23(1956), no. 3, 409–419.Google Scholar
Shelah, S., Proper and improper forcin . Second edition. Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. https://doi.org/10.1007/978-3-662-12831-2 Google Scholar
Tukey, J. W., Convergence and uniformity in topology . Annals of Mathematics Studies, 2, Princeton University Press, Princeton, NJ, 1940.Google Scholar
Wimmers, E. L., The Shelah P-point independence theorem . Israel J. Math. 43(1982), no. 1, 28–48. https://doi.org/10.1007/BF02761683 Google Scholar