Article
| Title: | On the bounding, splitting, and distributivity numbers (English) |
| Author: | Dow, Alan |
| Author: | Shelah, Saharon |
| Language: | English |
| Journal: | Commentationes Mathematicae Universitatis Carolinae |
| ISSN: | 0010-2628 (print) |
| ISSN: | 1213-7243 (online) |
| Volume: | 64 |
| Issue: | 3 |
| Year: | 2023 |
| Pages: | 331-351 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | The cardinal invariants $ \mathfrak h, \mathfrak b,\mathfrak s$ of $ \mathcal P (\omega)$ are known to satisfy that $\omega_1 \leq \mathfrak h \leq\min\{\mathfrak b, \mathfrak s\}$. We prove that all inequalities can be strict. We also introduce a new upper bound for $\mathfrak h$ and show that it can be less than $\mathfrak s$. The key method is to utilize finite support matrix iterations of ccc posets following paper Ultrafilters with small generating sets by A. Blass and S. Shelah (1989). (English) |
| Keyword: | cardinal invariants of the continuum |
| Keyword: | matrix forcing |
| MSC: | 03E15 |
| DOI: | 10.14712/1213-7243.2024.001 |
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| Date available: | 2024-03-18T10:43:37Z |
| Last updated: | 2024-03-18 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152302 |
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| Reference: | [1] Balcar B., Pelant J., Simon P.: The space of ultrafilters on $\mathbb N$ covered by nowhere dense sets.Fund. Math. 110 (1980), no. 1, pages 11–24. MR 0600576, 10.4064/fm-110-1-11-24 |
| Reference: | [2] Baumgartner J. E., Dordal P.: Adjoining dominating functions.J. Symbolic Logic 50 (1985), no. 1, 94–101. MR 0780528, 10.2307/2273792 |
| Reference: | [3] Blass A.: Applications of superperfect forcing and its relatives.Conf. Set Theory and Its Applications, Toronto, 1987, Lecture Notes in Math., 1401, Springer, Berlin, 1989, pages 18–40. MR 1031763, 10.1007/BFb0097329 |
| Reference: | [4] Blass A., Shelah S.: Ultrafilters with small generating sets.Israel J. Math. 65 (1989), no. 3, 259–271. MR 1005010, 10.1007/BF02764864 |
| Reference: | [5] Brendle J., Fischer V.: Mad families, splitting families and large continuum.J. Symbolic Logic 76 (2011), no. 1, 198–208. MR 2791343, 10.2178/jsl/1294170995 |
| Reference: | [6] Brendle J., Raghavan D.: Bounding, splitting, and almost disjointness.Ann. Pure Appl. Logic 165 (2014), no. 2, 631–651. MR 3129732, 10.1016/j.apal.2013.09.002 |
| Reference: | [7] Dow A., Shelah S.: On the cofinality of the splitting number.Indag. Math. (N.S.) 29 (2018), no. 1, 382–395. MR 3739621, 10.1016/j.indag.2017.01.010 |
| Reference: | [8] Dow A., Shelah S.: Pseudo P-points and splitting number.Arch. Math. Logic 58 (2019), no. 7–8, 1005–1027. MR 4003647, 10.1007/s00153-019-00674-x |
| Reference: | [9] Fischer V., Friedman S. D., Mejía D. A., Montoya D. C.: Coherent systems of finite support iterations.J. Symb. Log. 83 (2018), no. 1, 208–236. MR 3796283, 10.1017/jsl.2017.20 |
| Reference: | [10] Fischer V., Mejia D. A.: Splitting, bounding, and almost disjointness can be quite different.Canad. J. Math. 69 (2017), no. 3, 502–531. MR 3679685, 10.4153/CJM-2016-021-8 |
| Reference: | [11] Fischer V., Steprāns J.: The consistency of $ \mathfrak b=\kappa$ and $\mathfrak s=\kappa^+$.Fund. Math. 201 (2008), no. 3, 283–293. MR 2457482 |
| Reference: | [12] Goldstern M., Kellner J., Mejía D. A., Shelah S.: Preservation of splitting families and cardinal characteristics of the continuum.Israel J. Math. 246 (2021), no. 1, 73–129. MR 4358274, 10.1007/s11856-021-2237-7 |
| Reference: | [13] Ihoda J. I., Shelah S.: Souslin forcing.J. Symbolic Logic 53 (1988), no. 4, 1188–1207. MR 0973109, 10.2307/2274613 |
| Reference: | [14] Jech T.: Set Theory.Springer Monographs in Mathematics, Springer, Berlin, 2003. Zbl 1007.03002, MR 1940513 |
| Reference: | [15] Kunen K., Vaughan J. E.: Handbook of Set-theoretic Topology.North-Holland Publishing Co., Amsterdam, 1984. Zbl 0674.54001, MR 0776619 |
| Reference: | [16] Mejía D. A.: Matrix iterations and Cichon's diagram.Arch. Math. Logic 52 (2013), no. 3–4, 261–278. MR 3047455, 10.1007/s00153-012-0315-6 |
| Reference: | [17] Shelah S.: On cardinal invariants of the continuum.Conf. Axiomatic Set Theory, Boulder, 1983, Contemp. Math., 31, Amer. Math. Soc., Providence, 1984, pages 183–207. Zbl 0583.03035, MR 0763901, 10.1090/conm/031/763901 |
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