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  1. (1 other version)Ultrafilters on ω.James E. Baumgartner - 1995 - Journal of Symbolic Logic 60 (2):624-639.
    We study the I-ultrafilters on ω, where I is a collection of subsets of a set X, usually R or ω 1 . The I-ultrafilters usually contain the P-points, often as a small proper subset. We study relations between I-ultrafilters for various I, and closure of I-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether I-ultrafilters always exist.
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  • (1 other version)A few special ordinal ultrafilters.Claude Laflamme - 1996 - Journal of Symbolic Logic 61 (3):920-927.
    We prove various results on the notion of ordinal ultrafilters introduced by J. Baumgartner. In particular, we show that this notion of ultrafilter complexity is independent of the more familiar Rudin-Keisler ordering.
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  • P-hierarchy on β ω.Andrzej Starosolski - 2008 - Journal of Symbolic Logic 73 (4):1202-1214.
    We classify ultrafilters on ω with respect to sequential contours (see [4].[5]) of different ranks. In this way we obtain an ω1 sequence {Pα}1≤α≤ω1 of disjoint classes. We prove that non-emptiness of Pα for successor α ≥ 2 is equivalent to the existence of P-point. We investigate relations between P-hierarchy and ordinal ultrafilters (introduced by J. E. Baumgartner in [1]), we prove that it is relatively consistent with ZFC that the successor classes (for α ≥ 2) of P-hierarchy and ordinal (...)
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  • Cascades, order, and ultrafilters.Andrzej Starosolski - 2014 - Annals of Pure and Applied Logic 165 (10):1626-1638.
    We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. This allows us to prove that the class of strict JωωJωω-ultrafilters, introduced by J.E. Baumgartner in [2], is empty. We translate the result to the language of <∞<∞-sequences under an ultrafilter, investigated by C. Laflamme in [17], and we show that if there is an arbitrary long finite <∞<∞-sequence under u , then u is at least a strict Jωω+1Jωω+1-ultrafilter.
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  • How high can Baumgartner’s $${\mathcal{I}}$$ I -ultrafilters lie in the P-hierarchy?Michał Machura & Andrzej Starosolski - 2015 - Archive for Mathematical Logic 54 (5-6):555-569.
    Under the continuum hypothesis we prove that for any tall P-ideal Ionω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{I} \,{\rm on}\,\, \omega}$$\end{document} and for any ordinal γ≤ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\gamma \leq \omega_1}$$\end{document} there is an I\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{I}}$$\end{document}-ultrafilter in the sense of Baumgartner, which belongs to the class Pγ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{P}_{\gamma}}$$\end{document} of the P-hierarchy of ultrafilters. Since the class (...)
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  • (1 other version)Ultrafilters on $omega$.James E. Baumgartner - 1995 - Journal of Symbolic Logic 60 (2):624-639.
    We study the $I$-ultrafilters on $\omega$, where $I$ is a collection of subsets of a set $X$, usually $\mathbb{R}$ or $\omega_1$. The $I$-ultrafilters usually contain the $P$-points, often as a small proper subset. We study relations between $I$-ultrafilters for various $I$, and closure of $I$-ultrafilters under ultrafilter sums. We consider, but do not settle, the question whether $I$-ultrafilters always exist.
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