The near coherence of filters principle does not imply the filter dichotomy principle
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- by Heike Mildenberger and Saharon Shelah PDF
- Trans. Amer. Math. Soc. 361 (2009), 2305-2317 Request permission
Abstract:
We show that there is a forcing extension in which any two ultrafilters on $\omega$ are nearly coherent and there is a non-meagre filter that is not nearly ultra. This answers Blass’ longstanding question (1989) of whether the principle of near coherence of filters is strictly weaker than the filter dichotomy principle.References
- Tomek Bartoszyński and Haim Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR 1350295, DOI 10.1201/9781439863466
- Andreas Blass, Ultrafilters related to Hindman’s finite-unions theorem and its extensions, Logic and combinatorics (Arcata, Calif., 1985) Contemp. Math., vol. 65, Amer. Math. Soc., Providence, RI, 1987, pp. 89–124. MR 891244, DOI 10.1090/conm/065/891244
- Andreas Blass, Applications of superperfect forcing and its relatives, Set theory and its applications (Toronto, ON, 1987) Lecture Notes in Math., vol. 1401, Springer, Berlin, 1989, pp. 18–40. MR 1031763, DOI 10.1007/BFb0097329
- —, Combinatorial cardinal characteristics of the continuum, Handbook of Set Theory (Matthew Foreman, Akihiro Kanamori, and Menachem Magidor, eds.), Kluwer, to appear, available at http://www.math.lsa.umich.edu/$\sim$ablass.
- Andreas Blass and Claude Laflamme, Consistency results about filters and the number of inequivalent growth types, J. Symbolic Logic 54 (1989), no. 1, 50–56. MR 987321, DOI 10.2307/2275014
- Andreas Blass and Saharon Shelah, There may be simple $P_{\aleph _1}$- and $P_{\aleph _2}$-points and the Rudin-Keisler ordering may be downward directed, Ann. Pure Appl. Logic 33 (1987), no. 3, 213–243. MR 879489, DOI 10.1016/0168-0072(87)90082-0
- Andreas Blass and Saharon Shelah, Near coherence of filters. III. A simplified consistency proof, Notre Dame J. Formal Logic 30 (1989), no. 4, 530–538. MR 1036674, DOI 10.1305/ndjfl/1093635236
- Jörg Brendle, Distinguishing groupwise density numbers, Monatsh. Math. 152 (2007), no. 3, 207–215. MR 2357517, DOI 10.1007/s00605-007-0465-5
- Todd Eisworth, Forcing and stable ordered-union ultrafilters, J. Symbolic Logic 67 (2002), no. 1, 449–464. MR 1889561, DOI 10.2178/jsl/1190150054
- Neil Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11. MR 349574, DOI 10.1016/0097-3165(74)90023-5
- Jussi Ketonen, On the existence of $P$-points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), no. 2, 91–94. MR 433387, DOI 10.4064/fm-92-2-91-94
- Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam-New York, 1980. An introduction to independence proofs. MR 597342
- P. Matet, Partitions and filters, J. Symbolic Logic 51 (1986), no. 1, 12–21. MR 830067, DOI 10.2307/2273937
- Heike Mildenberger, Groupwise dense families, Arch. Math. Logic 40 (2001), no. 2, 93–112. MR 1816480, DOI 10.1007/s001530000049
- Heike Mildenberger, Saharon Shelah, and Boaz Tsaban, Covering the Baire space by families which are not finitely dominating, Ann. Pure Appl. Logic 140 (2006), no. 1-3, 60–71. MR 2224049, DOI 10.1016/j.apal.2005.09.008
- Arnold W. Miller, There are no $Q$-points in Laver’s model for the Borel conjecture, Proc. Amer. Math. Soc. 78 (1980), no. 1, 103–106. MR 548093, DOI 10.1090/S0002-9939-1980-0548093-2
- Andrzej Rosłanowski and Saharon Shelah, Norms on possibilities. I. Forcing with trees and creatures, Mem. Amer. Math. Soc. 141 (1999), no. 671, xii+167. MR 1613600, DOI 10.1090/memo/0671
- Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
- Michel Talagrand, Compacts de fonctions mesurables et filtres non mesurables, Studia Math. 67 (1980), no. 1, 13–43 (French). MR 579439, DOI 10.4064/sm-67-1-13-43
- Peter Vojtáš, Set-theoretic characteristics of summability of sequences and convergence of series, Comment. Math. Univ. Carolin. 28 (1987), no. 1, 173–183. MR 889779
Additional Information
- Heike Mildenberger
- Affiliation: Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Str. 25, 1090 Wien, Austria
- Email: heike@logic.univie.ac.at
- Saharon Shelah
- Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, 91904 Jerusalem, Israel – and – Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- Email: shelah@math.huji.ac.il
- Received by editor(s): November 16, 2006
- Published electronically: December 11, 2008
- Additional Notes: The first author was partially supported by the Landau Center.
The second author’s research was partially supported by the United States-Israel Binational Science Foundation (Grant no. 2002323). This is the second author’s publication no. 894. - © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 2305-2317
- MSC (2000): Primary 03E35, 03E17, 03E75, 54D40
- DOI: https://doi.org/10.1090/S0002-9947-08-04806-X
- MathSciNet review: 2471919
Dedicated: Dedicated to Andreas Blass on the occasion of his 60th birthday.