Abstract
Under the continuum hypothesis we prove that for any tall P-ideal \({\mathcal{I} \,{\rm on}\,\, \omega}\) and for any ordinal \({\gamma \leq \omega_1}\) there is an \({\mathcal{I}}\)-ultrafilter in the sense of Baumgartner, which belongs to the class \({\mathcal{P}_{\gamma}}\) of the P-hierarchy of ultrafilters. Since the class of \({\mathcal{P}_2}\) ultrafilters coincides with the class of P-points, our result generalizes the theorem of Flašková, which states that there are \({\mathcal{I}}\)-ultrafilters which are not P-points.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baumgartner J.E.: Ultrafilters on \({\omega}\). J. Symb. Log. 60(2), 624–639 (1995)
Błaszczyk A.: Free Boolean algebras and nowhere dense ultrafilters. Ann. Pure Appl. Log. 126, 287–292 (2004)
Brendle J.: P-points and nowhere dense ultrafilters. Isr. J. Math. 113, 205–230 (1999)
Daguenet M.: Emploi des filtres sur N dans l’étude descriptive des fonctions. Fundam. Math. 95, 11–33 (1977)
Dolecki S.: Multisequences. Quaest. Math. 29, 239–277 (2006)
Dolecki S., Mynard F.: Cascades and multifilters. Topol. Appl. 104, 53–65 (2002)
Dolecki S., Starosolski A., Watson S.: Extension of multisequences and countably uniradial classes of topologies. Comment. Math. Univ. Carolin. 44(1), 165–181 (2003)
Flašková J.: A note on I-ultrafilters and P-points. Acta Univ. Math. Carolin Math. Phys. 48(2), 43–48 (2007)
Flašková J.: More than a 0-point. Comment. Math. Univ. Carolin 47(4), 617–621 (2006)
Flašková J.: The relation of rapid ultrafilters and Q-points to van der Waerden ideal. Acta Univ. Carolin Math. Phys. 51(suppl.), 1927 (2010)
Flašková J.: Thin ultrafilters. Acta Univ. Carolin. Math. Phys. 46(2), 1319 (2005)
Flašková, J.: Ultrafilters and small sets. Doctoral thesis, Charles University in Prague, Faculty of Mathematics and Physics, Prague (2006)
Frolík Z.: Sums of ultrafilters. Bull. Am. Math. Soc. 73, 87–91 (1967)
Grimeisen G.: Gefilterte Summation von Filtern und iterierte Grenzprozesse, I. Math. Ann. 141, 318–342 (1960)
Grimeisen G.: Gefilterte Summation von Filtern und iterierte Grenzprozesse, II. Math. Ann. 144, 386–417 (1961)
Katětov M.: On Descriptive Classes of Functions Theory of Sets and Topology—A Collection of Papers in Honour of Felix Hausdorff. Deutscher Verlag der Wissenschaften, Berlin (1972)
Katětov, M.: On descriptive classification of functions. In: General Topology and Its Relations to Modern Analysis and Algebra II, Proceedings of Symposium, Prague (1971)
Laflamme C.: A few special ordinal ultrafilters. J. Symb. Log. 61(3), 920–927 (1996)
Shelah, S.: There may be no nowhere dense ultrafilters. In: Logic Colloquium Haifa’95, Lecture Notes Logic, vol. 11, pp. 305–324. Springer (1998); mathLO/9611221
Shelah S.: On what I do not understand (and have something to say): part I. Fundam. Math. 166, 1–82 (2000)
Starosolski A.: Fractalness of supercontours. Top. Proc. 30(1), 389–402 (2006)
Starosolski, A.: Cascades, order and ultrafilters. Ann. Pure Appl. Log. 165, 1626–1638 (2014)
Starosolski A.: P-hierarchy on \({\beta\omega}\). J. Symb. Log. 73(4), 1202–1214 (2008)
Starosolski A.: Ordinal ultrafilters versus P-hierarchy. Cent. Eur. J. Math. 12(1), 84–96 (2014)
Starosolski, A.: Topological aproach to ordinal ultrafilters and P-hierarchy. Preprint
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
About this article
Cite this article
Machura, M., Starosolski, A. How high can Baumgartner’s \({\mathcal{I}}\)-ultrafilters lie in the P-hierarchy?. Arch. Math. Logic 54, 555–569 (2015). https://doi.org/10.1007/s00153-015-0427-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-015-0427-x