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
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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.

References

Bohuslav Balcar and Petr Simon, On minimal $\pi$-character of points in extremally disconnected compact spaces, Topology Appl. 41 (1991), no. 1-2, 133–145. MR 1129703, DOI 10.1016/0166-8641(91)90105-U
Tomek Bartoszyński, On covering of real line by null sets, Pacific J. Math. 131 (1988), no. 1, 1–12. MR 917862, DOI 10.2140/pjm.1988.131.1
Tomek Bartoszyński and Haim Judah, Measure and category—filters on $\omega$, Set theory of the continuum (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 26, Springer, New York, 1992, pp. 175–201. MR 1233820, DOI 10.1007/978-1-4613-9754-0_{1}4
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
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, DOI 10.1017/CBO9780511758867.002
Murray G. Bell, On the combinatorial principle $P({\mathfrak {c}})$, Fund. Math. 114 (1981), no. 2, 149–157. MR 643555, DOI 10.4064/fm-114-2-149-157
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
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, DOI 10.1305/ndjfl/1093636772
Andreas Blass, Selective ultrafilters and homogeneity, Ann. Pure Appl. Logic 38 (1988), no. 3, 215–255. MR 942525, DOI 10.1016/0168-0072(88)90027-9
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
A. Blass and H. Mildenberger, On the cofinality of ultraproducts, preprint.
Andreas Blass and Saharon Shelah, Ultrafilters with small generating sets, Israel J. Math. 65 (1989), no. 3, 259–271. MR 1005010, DOI 10.1007/BF02764864
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
Jörg Brendle, Strolling through paradise, Fund. Math. 148 (1995), no. 1, 1–25. MR 1354935, DOI 10.4064/fm-148-1-1-25
Michael Canjar, Countable ultraproducts without CH, Ann. Pure Appl. Logic 37 (1988), no. 1, 1–79. MR 924678, DOI 10.1016/0168-0072(88)90048-6
R. Michael Canjar, Mathias forcing which does not add dominating reals, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1239–1248. MR 969054, DOI 10.1090/S0002-9939-1988-0969054-2
R. Michael Canjar, On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1, 233–241. MR 993747, DOI 10.1090/S0002-9939-1990-0993747-3
Keith J. Devlin, Constructibility, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1984. MR 750828, DOI 10.1007/978-3-662-21723-8
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, DOI 10.2307/2274354
Peter Lars Dordal, Towers in $[\omega ]^\omega$ and ${}^\omega \omega$, Ann. Pure Appl. Logic 45 (1989), no. 3, 247–276. MR 1032832, DOI 10.1016/0168-0072(89)90038-9
Erik Ellentuck, A new proof that analytic sets are Ramsey, J. Symbolic Logic 39 (1974), 163–165. MR 349393, DOI 10.2307/2272356
Martin Goldstern, Miroslav Repický, Saharon Shelah, and Otmar Spinas, On tree ideals, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1573–1581. MR 1233972, DOI 10.1090/S0002-9939-1995-1233972-4
M. Goldstern and S. Shelah, Ramsey ultrafilters and the reaping number—$\textrm {Con}({\mathfrak {r}}<{\mathfrak {u}})$, Ann. Pure Appl. Logic 49 (1990), no. 2, 121–142. MR 1077075, DOI 10.1016/0168-0072(90)90063-8
Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
T. Jech, Multiple forcing, Cambridge Tracts in Mathematics, vol. 88, Cambridge University Press, Cambridge, 1986. MR 895139
Haim Judah and Saharon Shelah, ${\bf \Delta }^1_3$-sets of reals, J. Symbolic Logic 58 (1993), no. 1, 72–80. MR 1217177, DOI 10.2307/2275325
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
Grzegorz Łabędzki and Miroslav Repický, Hechler reals, J. Symbolic Logic 60 (1995), no. 2, 444–458. MR 1335127, DOI 10.2307/2275841
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, DOI 10.1007/BF02756789
J. C. Oxtoby and S. M. Ulam, On the existence of a measure invariant under a transformation, Ann. of Math. (2) 40 (1939), 560–566. MR 97, DOI 10.2307/1968940
A. R. D. Mathias, Happy families, Ann. Math. Logic 12 (1977), no. 1, 59–111. MR 491197, DOI 10.1016/0003-4843(77)90006-7
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, DOI 10.1090/S0002-9939-1982-0671224-2
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
P. Nyikos, Special ultrafilters and cofinal subsets of $\omega ^\omega$, preprint.
Szymon Plewik, On completely Ramsey sets, Fund. Math. 127 (1987), no. 2, 127–132. MR 882622, DOI 10.4064/fm-127-2-127-132
Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955, DOI 10.1007/978-3-662-21543-2
Saharon Shelah and Juris Steprāns, Erratum: “Maximal chains in ${}^\omega \omega$ 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, DOI 10.1007/BF01352936
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
Eric K. van Douwen, The integers and topology, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 111–167. MR 776622
Jan van Mill, An introduction to $\beta \omega$, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 503–567. MR 776630
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.