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The near coherence of filters principle does not imply the filter dichotomy principle
Authors:
Heike Mildenberger and Saharon Shelah
Journal:
Trans. Amer. Math. Soc. 361 (2009), 2305-2317
MSC (2000):
Primary 03E35, 03E17, 03E75, 54D40
Posted:
December 11, 2008
MathSciNet review:
2471919
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Additional Information
Abstract: We show that there is a forcing extension in which any two ultrafilters on  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.
- 1.
Tomek
Bartoszyński and Haim
Judah, Set theory, A K Peters Ltd., Wellesley, MA, 1995. On
the structure of the real line. MR 1350295
(96k:03002)
- 2.
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
(88g:04002), http://dx.doi.org/10.1090/conm/065/891244
- 3.
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
(91b:03081), http://dx.doi.org/10.1007/BFb0097329
- 4.
-, 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/
 ablass.
- 5.
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
(90a:03076), http://dx.doi.org/10.2307/2275014
- 6.
Andreas
Blass and Saharon
Shelah, There may be simple 𝑃_{ℵ₁}- and
𝑃_{ℵ₂}-points and the Rudin-Keisler ordering may be
downward directed, Ann. Pure Appl. Logic 33 (1987),
no. 3, 213–243. MR 879489
(88e:03073), http://dx.doi.org/10.1016/0168-0072(87)90082-0
- 7.
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
(90m:03087), http://dx.doi.org/10.1305/ndjfl/1093635236
- 8.
Jörg
Brendle, Distinguishing groupwise density numbers, Monatsh.
Math. 152 (2007), no. 3, 207–215. MR 2357517
(2009c:03046), http://dx.doi.org/10.1007/s00605-007-0465-5
- 9.
Todd
Eisworth, Forcing and stable ordered-union ultrafilters, J.
Symbolic Logic 67 (2002), no. 1, 449–464. MR 1889561
(2003d:03074), http://dx.doi.org/10.2178/jsl/1190150054
- 10.
Neil
Hindman, Finite sums from sequences within cells of a partition of
𝑁, J. Combinatorial Theory Ser. A 17 (1974),
1–11. MR
0349574 (50 #2067)
- 11.
Jussi
Ketonen, On the existence of 𝑃-points in the
Stone-Čech compactification of integers, Fund. Math.
92 (1976), no. 2, 91–94. MR 0433387
(55 #6363)
- 12.
Kenneth
Kunen, Set theory, Studies in Logic and the Foundations of
Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1980.
An introduction to independence proofs. MR 597342
(82f:03001)
- 13.
P.
Matet, Partitions and filters, J. Symbolic Logic
51 (1986), no. 1, 12–21. MR 830067
(87g:03052), http://dx.doi.org/10.2307/2273937
- 14.
Heike
Mildenberger, Groupwise dense families, Arch. Math. Logic
40 (2001), no. 2, 93–112. MR 1816480
(2003d:03077), http://dx.doi.org/10.1007/s001530000049
- 15.
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
(2007k:03120), http://dx.doi.org/10.1016/j.apal.2005.09.008
- 16.
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
(80h:03071), http://dx.doi.org/10.1090/S0002-9939-1980-0548093-2
- 17.
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
(2000c:03036)
- 18.
Saharon
Shelah, Proper and improper forcing, 2nd ed., Perspectives in
Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206
(98m:03002)
- 19.
Michel
Talagrand, Compacts de fonctions mesurables et filtres non
mesurables, Studia Math. 67 (1980), no. 1,
13–43 (French). MR 579439
(82e:28009)
- 20.
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
(88i:40001)
- 1.
- Tomek Bartoszyński and Haim Judah, Set Theory, On the Structure of the Real Line, A. K. Peters, 1995. MR 1350295 (96k:03002)
- 2.
- Andreas Blass, Ultrafilters related to Hindman's finite-unions theorem and its extensions, Logic and Combinatorics (S. Simpson, ed.), Contemp. Math., vol. 65, Amer. Math. Soc., 1987, pp. 89-124. MR 891244 (88g:04002)
- 3.
- -, Applications of superperfect forcing and its relatives, Set Theory and its Applications (Juris Steprāns and Steve Watson, eds.), Lecture Notes in Mathematics, vol. 1401, 1989, pp. 18-40. MR 1031763 (91b:03081)
- 4.
- -, 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/ablass.
- 5.
- Andreas Blass and Claude Laflamme, Consistency results about filters and the number of inequivalent growth types, J. Symbolic Logic 54 (1989), 50-56. MR 987321 (90a:03076)
- 6.
- Andreas Blass and Saharon Shelah, There may be simple
 - and  -points and the Rudin-Keisler ordering may be downward directed, Annals of Pure and Applied Logic 33 (1987), 213-243. MR 879489 (88e:03073)
- 7.
- -, Near coherence of filters. III. A simplified consistency proof, Notre Dame Journal of Formal Logic 30 (1989), 530-538. MR 1036674 (90m:03087)
- 8.
- Jörg Brendle, Distinguishing groupwise density numbers, Monatshefte für Mathematik 152 (2007), no. 3, 207-215. MR 2357517
- 9.
- Todd Eisworth, Forcing and stable ordered-union ultrafilters, J. Symbolic Logic 67 (2002), 449-464. MR 1889561 (2003d:03074)
- 10.
- Neil Hindman, Finite sums from sequences within cells of a partition of
 , J. Combin. Theory Ser. A 17 (1974), 1-11. MR 0349574 (50:2067)
- 11.
- Jussi Ketonen, On the existence of
 -points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91-94. MR 0433387 (55:6363)
- 12.
- Kenneth Kunen, Set theory, an introduction to independence proofs, North-Holland, 1980. MR 597342 (82f:03001)
- 13.
- Pierre Matet, Partitions and filters, J. Symbolic Logic 51 (1986), 12-21. MR 830067 (87g:03052)
- 14.
- Heike Mildenberger, Groupwise dense families, Arch. Math. Logic 40 (2000), 93-112. MR 1816480 (2003d:03077)
- 15.
- Heike Mildenberger, Saharon Shelah, and Boaz Tsaban, Covering the Baire space by families which are not finitely dominating, Annals of Pure and Applied Logic 140 (2006), 60-71. MR 2224049 (2007k:03120)
- 16.
- Arnold Miller, There are no
 -points in Laver's model for the Borel conjecture, Proc. Amer. Math. Soc. 78 (1980), 103-106. MR 548093 (80h:03071)
- 17.
- Andrzej Rosłanowski and Saharon Shelah, Norms on Possibilities I: Forcing with Trees and Creatures, Memoirs of the American Mathematical Society, vol. 141 (no. 671), AMS, 1999. MR 1613600 (2000c:03036)
- 18.
- Saharon Shelah, Proper and Improper Forcing, 2nd Edition, Springer, 1998. MR 1623206 (98m:03002)
- 19.
- Michel Talagrand, Compacts de fonctions mesurables et filtres non mesurables, Studia Mathematicae 67 (1980), 13-43. MR 579439 (82e:28009)
- 20.
- Peter Vojtáš, Set theoretic characteristics of summability and convergence of series, Commentationes Mathematicae Universitatis Carolinae 28 (1987), 173-184. MR 889779 (88i:40001)
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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
Email:
shelah@math.huji.ac.il
DOI:
http://dx.doi.org/10.1090/S0002-9947-08-04806-X
PII:
S 0002-9947(08)04806-X
Received by editor(s):
November 16, 2006
Posted:
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.
Dedicated:
Dedicated to Andreas Blass on the occasion of his 60th birthday.
Article copyright:
© Copyright 2008 American Mathematical Society
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