Partitioning a reflecting stationary set
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- by Maxwell Levine and Assaf Rinot PDF
- Proc. Amer. Math. Soc. 148 (2020), 3551-3565 Request permission
Abstract:
We address the question of whether a reflecting stationary set may be partitioned into two or more reflecting stationary subsets, providing various affirmative answers in ZFC. As an application to singular cardinal combinatorics, we infer that it is never the case that there exists a singular cardinal all of whose scales are very good.References
- Ari Meir Brodsky and Assaf Rinot, Distributive Aronszajn trees, Fund. Math. 245 (2019), no. 3, 217–291. MR 3914943, DOI 10.4064/fm542-4-2018
- James Cummings, Mirna D amonja, and Saharon Shelah, A consistency result on weak reflection, Fund. Math. 148 (1995), no. 1, 91–100. MR 1354940, DOI 10.4064/fm-148-1-91-100
- James Cummings and Matthew Foreman, Diagonal Prikry extensions, J. Symbolic Logic 75 (2010), no. 4, 1383–1402. MR 2767975, DOI 10.2178/jsl/1286198153
- Todd Eisworth, Successors of singular cardinals, Handbook of set theory. Vols. 1, 2, 3, Springer, Dordrecht, 2010, pp. 1229–1350. MR 2768694, DOI 10.1007/978-1-4020-5764-9_{1}6
- Moti Gitik and Assaf Sharon, On SCH and the approachability property, Proc. Amer. Math. Soc. 136 (2008), no. 1, 311–320. MR 2350418, DOI 10.1090/S0002-9939-07-08716-3
- Menachem Kojman, On the arithmetic of density, Topology Appl. 213 (2016), 145–153. MR 3563076, DOI 10.1016/j.topol.2016.08.016
- Paul Larson and Saharon Shelah, Splitting stationary sets from weak forms of choice, MLQ Math. Log. Q. 55 (2009), no. 3, 299–306. MR 2519245, DOI 10.1002/malq.200810011
- Menachem Magidor, Reflecting stationary sets, J. Symbolic Logic 47 (1982), no. 4, 755–771 (1983). MR 683153, DOI 10.2307/2273097
- Assaf Rinot, Chain conditions of products, and weakly compact cardinals, Bull. Symb. Log. 20 (2014), no. 3, 293–314. MR 3271280, DOI 10.1017/bsl.2014.24
- Saharon Shelah, Reflecting stationary sets and successors of singular cardinals, Arch. Math. Logic 31 (1991), no. 1, 25–53. MR 1126352, DOI 10.1007/BF01370693
- Saharon Shelah, Advances in cardinal arithmetic, Finite and infinite combinatorics in sets and logic (Banff, AB, 1991) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 411, Kluwer Acad. Publ., Dordrecht, 1993, pp. 355–383. MR 1261217
- Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912
- Robert M. Solovay, Real-valued measurable cardinals, Axiomatic set theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967) Amer. Math. Soc., Providence, R.I., 1971, pp. 397–428. MR 0290961
- Assaf Sharon and Matteo Viale, Some consequences of reflection on the approachability ideal, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4201–4212. MR 2608402, DOI 10.1090/S0002-9947-10-04976-7
- Stanisław Marcin Ulam, Zur masstheorie in der allgemeinen mengenlehre, Fund. Math., 16 (1930), no 1, 140–150.
- W. Hugh Woodin, The axiom of determinacy, forcing axioms, and the nonstationary ideal, Second revised edition, De Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter GmbH & Co. KG, Berlin, 2010. MR 2723878, DOI 10.1515/9783110213171
Additional Information
- Maxwell Levine
- Affiliation: Universität Wien, Kurt Gödel Research Center for Mathematical Logic, Wien, Austria
- MR Author ID: 1266425
- ORCID: 0000-0001-7150-102X
- Email: maxwell.levine@univie.ac.at
- Assaf Rinot
- Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel
- MR Author ID: 785097
- Email: rinotas@math.biu.ac.il
- Received by editor(s): February 15, 2019
- Received by editor(s) in revised form: July 19, 2019
- Published electronically: May 4, 2020
- Additional Notes: The second author was partially supported by the European Research Council (grant agreement ERC-2018-StG 802756) and by the Israel Science Foundation (grant agreement 2066/18).
- Communicated by: Heike Mildenberger
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3551-3565
- MSC (2010): Primary 03E05; Secondary 03E04
- DOI: https://doi.org/10.1090/proc/14783
- MathSciNet review: 4108860