Preservation of splitting families and cardinal characteristics of the continuum. (English) Zbl 07513387
Summary: We show how to construct, via forcing, splitting families that are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń’s diagram, \( \mathfrak{m}\)(2-Knaster), \( \mathfrak{p}\), \(\mathfrak{h}\), the splitting number \(\mathfrak{s}\) and the reaping number \(\mathfrak{r}\).
References:
| [1] | J. Brendle, M. A. Cardona and D. A. Mejía, Filter-linkedness and its effect on preservation of cardinal characteristics, Annals of Pure and Applied Logic 171 (2021) Article no. 102856. · Zbl 1498.03108 |
| [2] | Baumgartner, J. E.; Dordal, P., Adjoining dominating functions, Journal of Symbolic Logic, 50, 94-101 (1985) · Zbl 0566.03031 · doi:10.2307/2273792 |
| [3] | Bell, M. G., On the combinatorial principle P(c), Fundamenta Mathematicae, 114, 149-157 (1981) · Zbl 0581.03038 · doi:10.4064/fm-114-2-149-157 |
| [4] | Brendle, J.; Fischer, V., Mad families, splitting families and large continuum, Journal of Symbolic Logic, 76, 198-208 (2011) · Zbl 1215.03061 · doi:10.2178/jsl/1294170995 |
| [5] | Balcar, B.; Hernández-Hernández, F.; Hrušák, M., Combinatorics of dense subsets of the rationals, Fundamenta Mathematicae, 183, 59-80 (2004) · Zbl 1051.03038 · doi:10.4064/fm183-1-4 |
| [6] | Bartoszyński, T.; Judah, H., Set Theory: On the Structure of the Real Line (1995), Wellesley, MA: A. K. Peters, Wellesley, MA · Zbl 0834.04001 · doi:10.1201/9781439863466 |
| [7] | Blass, A., Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory. Vols. 1, 2, 3, 395-489 (2010), Dordrecht: Springer, Dordrecht · Zbl 1198.03058 · doi:10.1007/978-1-4020-5764-9_7 |
| [8] | Brendle, J., Larger cardinals in Cichoń’s diagram, Journal of Symbolic Logic, 56, 795-810 (1991) · Zbl 0758.03021 |
| [9] | Dow, A.; Shelah, S., On the cofinality of the splitting number, Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae, 29, 382-395 (2018) · Zbl 1436.03260 |
| [10] | Engelking, R.; Karłowicz, M., Some theorems of set theory and their topological consequences, Fundamenta Mathematicae, 57, 275-285 (1965) · Zbl 0137.41904 · doi:10.4064/fm-57-3-275-285 |
| [11] | Fischer, V.; Friedman, S. D.; Mejía, D. A.; Montoya, D. C., Coherent systems of finite support iterations, Journal of Symbolic Logic, 83, 208-236 (2018) · Zbl 1447.03013 · doi:10.1017/jsl.2017.20 |
| [12] | Fischer, A.; Goldstern, M.; Kellner, J.; Shelah, S., Creature forcing and five cardinal characteristics in Cichoń’s diagram, Archive for Mathematical Logic, 56, 1045-1103 (2017) · Zbl 1404.03040 · doi:10.1007/s00153-017-0553-8 |
| [13] | M. Goldstern, J. Kellner, D. A. Mejía and S. Shelah, Cichoń’s maximum without large cardinals, Journal of the European Mathematical Society, to appear, https://arxiv.org/abs/1906.06608. · Zbl 1493.03008 |
| [14] | M. Goldstern, J. Kellner, D. A. Mejía and S. Shelah, Controlling classical cardinal characteristics without adding reals, Journal of Mathematical Logic, doi:10.1142/S0219061321500185. · Zbl 07419668 |
| [15] | Goldstern, M.; Kellner, J.; Shelah, S., Cichoń’s maximum, Annals of Mathematics, 190, 113-143 (2019) · Zbl 1493.03008 · doi:10.4007/annals.2019.190.1.2 |
| [16] | Goldstern, M.; Mejía, D. A.; Shelah, S., The left side of Cichoń’s diagram, Proceedings of the American Mathematical Society, 144, 4025-4042 (2016) · Zbl 1431.03064 · doi:10.1090/proc/13161 |
| [17] | Hechler, S. H., Short complete nested sequences in βNN and small maximal almost-disjoint families, General Topology and its Applications, 2, 139-149 (1972) · Zbl 0246.02047 · doi:10.1016/0016-660X(72)90001-3 |
| [18] | Judah, H.; Shelah, S., Souslin forcing, Journal of Symbolic Logic, 53, 1188-1207 (1988) · Zbl 0673.03039 · doi:10.1017/S0022481200028012 |
| [19] | Judah, H.; Shelah, S., The Kunen-Miller chart (Lebesgue measure, the Baire property, Laver reals and preservation theorems for forcing, Journal of Symbolic Logic, 55, 909-927 (1990) · Zbl 0718.03037 · doi:10.2307/2274464 |
| [20] | Kamburelis, A., Iterations of Boolean algebras with measure, Archive for Mathematical Logic, 29, 21-28 (1989) · Zbl 0687.03032 · doi:10.1007/BF01630808 |
| [21] | Kellner, J.; Shelah, S., Decisive creatures and large continuum, Journal of Symbolic Logic, 74, 73-104 (2009) · Zbl 1183.03035 · doi:10.2178/jsl/1231082303 |
| [22] | Kellner, J.; Shelah, S., Creature forcing and large continuum: the joy of halving, Archive for Mathematical Logic, 51, 49-70 (2012) · Zbl 1259.03063 · doi:10.1007/s00153-011-0253-8 |
| [23] | Kellner, J.; Shelah, S.; Tănasie, A., Another ordering of the ten cardinal characteristics in cichońn’s diagram, Commentationes Mathematicae Universitatis Carolinae, 60, 61-95 (2019) · Zbl 1463.03015 · doi:10.14712/1213-7243.2015.273 |
| [24] | Kunen, K., Set Theory (1980), Amsterdam-New York: North-Holland, Amsterdam-New York · Zbl 0443.03021 |
| [25] | Kamburelis, A.; Weglorz, B., Splittings, Archive for Mathematical Logic, 35, 263-277 (1996) · Zbl 0852.04004 · doi:10.1007/s001530050044 |
| [26] | Mejía, D. A., Models of some cardinal invariants with large continuum, Kyōto Daigaku Sūrikaiseki Kenkyūsho Kōkyūroku, 1851, 36-48 (2013) |
| [27] | Mejía, D. A., Matrix iterations with vertical support restrictions, Proceedings of the 14th and 15th Asian Logic Conferences, 213-248 (2019), Hackensack, NJ: World Scientific, Hackensack, NJ |
| [28] | Mejía, D. A., A note on “Another ordering of the ten cardinal characteristics in Cichoń’s Diagram” and further remarks, Kyōto Daigaku Sūrikaiseki Kenkyūsho Kōokyūroku, 2141, 1-15 (2019) |
| [29] | Malliaris, M.; Shelah, S., Cofinality spectrum theorems in model theory, set theory, and general topology, Journal of the American Mathematical Society, 29, 237-297 (2016) · Zbl 1477.03125 · doi:10.1090/jams830 |
| [30] | Shelah, S., On cardinal invariants of the continuum, Axiomatic Set Theory (Boulder, Colo., 1983), 183-207 (1984), Providence, RI: American Mathematical Society, Providence, RI · Zbl 0583.03035 · doi:10.1090/conm/031/763901 |
| [31] | Shelah, S., Covering of the null ideal may have countable cofinality, Fundamenta Mathematicae, 166, 109-136 (2000) · Zbl 0962.03046 |
| [32] | Steprāns, J., Combinatorial consequences of adding Cohen reals, Set Theory of the Reals (Ramat Gan, 1991), 583-617 (1993), Ramat Gan: Bar-Ilan University, Ramat Gan · Zbl 0839.03037 |
| [33] | Vojtáš, P., Generalized Galois-Tukey-connections between explicit relations on classical objects of real analysis, Set Theory of the Reals (Ramat Gan, 1991), 619-643 (1993), Ramat Gan: Bar-Ilan University, Ramat Gan · Zbl 0829.03027 |
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