Ultrafilters on 𝜔-their ideals and their cardinal characteristics

Shelah, Saharon; Brendle, Jörg; Shelah, Saharon

doi:10.1090/S0002-9947-99-02257-6

Ultrafilters on $\omega$ -their ideals and their cardinal characteristics

Authors: Saharon Shelah, Jörg Brendle and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 351 (1999), 2643-2674
MSC (1991): Primary 03E05, 03E35
DOI: https://doi.org/10.1090/S0002-9947-99-02257-6
Published electronically: March 8, 1999
MathSciNet review: 1686797
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a free ultrafilter $\mathcal{U}$ on $\omega$ we study several cardinal characteristics which describe part of the combinatorial structure of $\,\mathcal{U}$ . We provide various consistency results; e.g. we show how to force simultaneously many characters and many $\pi$ -characters. We also investigate two ideals on the Baire space $\omega ^{\omega }$ naturally related to $\mathcal{U}$ and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.

[BS] Bohuslav Balcar and Petr Simon, On minimal 𝜋-character of points in extremally disconnected compact spaces, Topology Appl. 41 (1991), no. 1-2, 133–145. MR 1129703, https://doi.org/10.1016/0166-8641(91)90105-U
[Ba] Tomek Bartoszyński, On covering of real line by null sets, Pacific J. Math. 131 (1988), no. 1, 1–12. MR 917862
[BJ] Tomek Bartoszyński and Haim Judah, Measure and category—filters on 𝜔, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 175–201. MR 1233820, https://doi.org/10.1007/978-1-4613-9754-0_14
[BJ 1] Tomek Bartoszyński and Haim Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR 1350295
[B] James E. Baumgartner, Iterated forcing, Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87, Cambridge Univ. Press, Cambridge, 1983, pp. 1–59. MR 823775, https://doi.org/10.1017/CBO9780511758867.002
[Be] Murray G. Bell, On the combinatorial principle 𝑃(𝔠), Fund. Math. 114 (1981), no. 2, 149–157. MR 643555, https://doi.org/10.4064/fm-114-2-149-157
[BK] Murray Bell and Kenneth Kunen, On the PI character of ultrafilters, C. R. Math. Rep. Acad. Sci. Canada 3 (1981), no. 6, 351–356. MR 642449
[Bl] Andreas Blass, Near coherence of filters. I. Cofinal equivalence of models of arithmetic, Notre Dame J. Formal Logic 27 (1986), no. 4, 579–591. MR 867002, https://doi.org/10.1305/ndjfl/1093636772
[Bl 1] Andreas Blass, Selective ultrafilters and homogeneity, Ann. Pure Appl. Logic 38 (1988), no. 3, 215–255. MR 942525, https://doi.org/10.1016/0168-0072(88)90027-9
[Bl 2] Andreas Blass, Simple cardinal characteristics of the continuum, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 63–90. MR 1234278
[BlM] A. Blass and H. Mildenberger, On the cofinality of ultraproducts, preprint.
[BlS] Andreas Blass and Saharon Shelah, Ultrafilters with small generating sets, Israel J. Math. 65 (1989), no. 3, 259–271. MR 1005010, https://doi.org/10.1007/BF02764864
[BlS 1] 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, https://doi.org/10.1016/0168-0072(87)90082-0
[Br] Jörg Brendle, Strolling through paradise, Fund. Math. 148 (1995), no. 1, 1–25. MR 1354935, https://doi.org/10.4064/fm-148-1-1-25
[Ca] Michael Canjar, Countable ultraproducts without CH, Ann. Pure Appl. Logic 37 (1988), no. 1, 1–79. MR 924678, https://doi.org/10.1016/0168-0072(88)90048-6
[Ca 1] R. Michael Canjar, Mathias forcing which does not add dominating reals, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1239–1248. MR 969054, https://doi.org/10.1090/S0002-9939-1988-0969054-2
[Ca 2] R. Michael Canjar, On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1, 233–241. MR 993747, https://doi.org/10.1090/S0002-9939-1990-0993747-3
[De] Keith J. Devlin, Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1984. MR 750828
[Do] Peter Lars Dordal, A model in which the base-matrix tree cannot have cofinal branches, J. Symbolic Logic 52 (1987), no. 3, 651–664. MR 902981, https://doi.org/10.2307/2274354
[Do 1] Peter Lars Dordal, Towers in [𝜔]^{𝜔} and ^{𝜔}𝜔, Ann. Pure Appl. Logic 45 (1989), no. 3, 247–276. MR 1032832, https://doi.org/10.1016/0168-0072(89)90038-9
[El] Erik Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163–165. MR 349393, https://doi.org/10.2307/2272356
[GRSS] Martin Goldstern, Miroslav Repický, Saharon Shelah, and Otmar Spinas, On tree ideals, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1573–1581. MR 1233972, https://doi.org/10.1090/S0002-9939-1995-1233972-4
[GS] M. Goldstern and S. Shelah, Ramsey ultrafilters and the reaping number—𝐶𝑜𝑛(𝔯<𝔲), Ann. Pure Appl. Logic 49 (1990), no. 2, 121–142. MR 1077075, https://doi.org/10.1016/0168-0072(90)90063-8
[Je] Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. MR 506523
[Je 1] T. Jech, Multiple forcing, Cambridge Tracts in Mathematics, vol. 88, Cambridge University Press, Cambridge, 1986. MR 895139
[JS] Haim Judah and Saharon Shelah, Δ¹₃-sets of reals, J. Symbolic Logic 58 (1993), no. 1, 72–80. MR 1217177, https://doi.org/10.2307/2275325
[Ku] Kenneth Kunen, Set theory, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland Publishing Co., Amsterdam, 1983. An introduction to independence proofs; Reprint of the 1980 original. MR 756630
[LR] Grzegorz Łabędzki and Miroslav Repický, Hechler reals, J. Symbolic Logic 60 (1995), no. 2, 444–458. MR 1335127, https://doi.org/10.2307/2275841
[Lo] Alain Louveau, Une méthode topologique pour l’étude de la propriété de Ramsey, Israel J. Math. 23 (1976), no. 2, 97–116 (French, with English summary). MR 411971, https://doi.org/10.1007/BF02756789
[M] P. Matet, Combinatorics and forcing with distributive ideals, Annals of Pure and Applied Logic, vol. 86 (1997), pp. 2643-2674. MR 97:14
[Ma] A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59–111. MR 491197, https://doi.org/10.1016/0003-4843(77)90006-7
[Mi] Arnold W. Miller, A characterization of the least cardinal for which the Baire category theorem fails, Proc. Amer. Math. Soc. 86 (1982), no. 3, 498–502. MR 671224, https://doi.org/10.1090/S0002-9939-1982-0671224-2
[Mi 1] Arnold W. Miller, Arnie Miller’s problem list, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 645–654. MR 1234292
[Ny] P. Nyikos, Special ultrafilters and cofinal subsets of ${\omega ^\omega }$ , preprint.
[Pl] Szymon Plewik, On completely Ramsey sets, Fund. Math. 127 (1987), no. 2, 127–132. MR 882622, https://doi.org/10.4064/fm-127-2-127-132
[Sh] Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955
[SS] Saharon Shelah and Juris Steprāns, Erratum: “Maximal chains in ^{𝜔}𝜔 and ultrapowers of the integers” [Arch. Math. Logic 32 (1993), no. 5, 305–319; MR1223393 (94g:03094)], Arch. Math. Logic 33 (1994), no. 2, 167–168. MR 1271434, https://doi.org/10.1007/BF01352936
Saharon Shelah and Juris Steprāns, Maximal chains in ^{𝜔}𝜔 and ultrapowers of the integers, Arch. Math. Logic 32 (1993), no. 5, 305–319. MR 1223393, https://doi.org/10.1007/BF01409965
[St] Juris Steprāns, Combinatorial consequences of adding Cohen reals, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 583–617. MR 1234290
[vD] Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
[vM] Jan van Mill, An introduction to 𝛽𝜔, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 503–567. MR 776630
[Va] J. Vaughan, Small uncountable cardinals and topology, in: Open problems in topology (J. van Mill and G. Reed, eds.), North-Holland, 1990, pp. 195-218. CMP 91:03