Starosolski, Andrzej Cascades, order, and ultrafilters. (English) Zbl 1354.03061 Ann. Pure Appl. Logic 165, No. 10, 1626-1638 (2014). Summary: We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. This allows us to prove (ZFC) that the class of strict \(J_{\omega^\omega}\)-ultrafilters, introduced by J. E. Baumgartner [J. Symb. Log. 60, No. 2, 624–639 (1995; Zbl 0834.04005)], is empty. We translate the result to the language of \(<_\infty\)-sequences under an ultrafilter, investigated by C. Laflamme [J. Symb. Log. 61, No. 3, 920–927 (1996; Zbl 0871.04004)], and we show that if there is an arbitrary long finite \(<_\infty\)-sequence under \(u\), then \(u\) is at least a strict \(J_{\omega^{\omega+1}}\)-ultrafilter. Cited in 3 Documents MSC: 03E04 Ordered sets and their cofinalities; pcf theory 03E05 Other combinatorial set theory 03E20 Other classical set theory (including functions, relations, and set algebra) 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.) Keywords:ordinal ultrafilter; monotone sequential cascade Citations:Zbl 0834.04005; Zbl 0871.04004 PDFBibTeX XMLCite \textit{A. Starosolski}, Ann. Pure Appl. Logic 165, No. 10, 1626--1638 (2014; Zbl 1354.03061) Full Text: DOI arXiv References: [1] Barney, C., Ultrafilters on the natural numbers, J. Symbolic Logic, 68, 3, 764-784 (2003) · Zbl 1058.03046 [2] Baumgartner, J. E., Ultrafilters on \(ω\), J. Symbolic Logic, 60, 2, 624-639 (1995) · Zbl 0834.04005 [3] Błaszczyk, A., Free Boolean algebras and nowhere dense ultrafilters, Ann. Pure Appl. Logic, 126, 287-292 (2004) · Zbl 1054.03034 [4] Brendle, J., Between P-points and nowhere dense ultrafilters, Israel J. Math., 113, 205-230 (1999) · Zbl 0938.03069 [5] Daguenet, M., Emploi des filtres sur N dans l’étude descriptive des fonctions, Fund. Math., 95, 11-33 (1977) · Zbl 0362.04006 [6] Dolecki, S., Multisequences, Quaest. Math., 29, 239-277 (2006) · Zbl 1120.54002 [7] Dolecki, S.; Mynard, F., Cascades and multifilters, Topology Appl., 104, 53-65 (2002) · Zbl 0953.54003 [8] Dolecki, S.; Starosolski, A.; Watson, S., Extension of multisequences and countable uniradial class of topologies, Comment. Math. Univ. Carolin., 44, 1, 165-181 (2003) · Zbl 1099.54024 [9] Flašková, J., Thin ultrafilters, Acta Univ. Carolin. Math. Phys., 46, 2, 13-19 (2005) [10] Flašková, J., Ultrafilters and small sets (2006), Charles University: Charles University Prague, Ph.D. Thesis · Zbl 1204.03046 [11] Flašková, J., I-ultrafilters and summable ideals, (Proceedings of the 10th Asian Logic Conference. Proceedings of the 10th Asian Logic Conference, Kobe, 2008 (2010), World Scientific: World Scientific Singapore), 113-123 · Zbl 1204.03046 [12] Frolík, Z., Sums of ultrafilters, Bull. Amer. Math. Soc., 73, 87-91 (1967) · Zbl 0166.18602 [13] Grimeisen, G., Gefilterte Summation von Filtern und iterierte Grenzprozesse, I, Math. Ann., 141, 318-342 (1960) · Zbl 0096.26201 [14] Grimeisen, G., Gefilterte Summation von Filtern und iterierte Grenzprozesse, II, Math. Ann., 144, 386-417 (1961) · Zbl 0101.14805 [15] Katětov, M., On descriptive classification of functions, (General Topology and Its Relations to Modern Analysis and Algebra II. General Topology and Its Relations to Modern Analysis and Algebra II, Proc. Sympos., Prague (1971)) [16] Katětov, M., On descriptive classes of functions, (Theory of Sets and Topology—A Collection of Papers in Honour of Felix Hausdorff (1972), D. V. W.) · Zbl 0265.54014 [17] Kunen, K.; Vaughan, J. E., Handbook of Set-Theoretic Topology (1988), North-Holland · Zbl 0674.54001 [18] Shelah, S., There may be no nowhere dense ultrafilters, (Logic Colloquium Haifa’95. Logic Colloquium Haifa’95, Lect. Notes Log., vol. 11 (1998), Springer), 305-324 · Zbl 0894.03030 [19] Shelah, S., On what I do not understand (and have something to say), Fund. Math., 166, 1-82 (2000) · Zbl 0966.03044 [20] Starosolski, A., Fractalness of supercontours, Topology Proc., 30, 1, 389-402 (2006) · Zbl 1131.54020 [21] Starosolski, A., P-hierarchy on βω, J. Symbolic Logic, 73, 4, 1202-1214 (2008) · Zbl 1156.03047 [22] Starosolski, A., Ordinal ultrafilters versus P-hierarchy, Cent. Eur. J. Math., 12, 1, 84-96 (2014) · Zbl 1326.03055 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.