Abstract
We study the intersection number of families of tall ideals. We show that the intersection number of the class of analytic P-ideals is equal to the bounding number \({\mathfrak{b}}\), the intersection number of the class of all meager ideals is equal to \({\mathfrak{h}}\) and the intersection number of the class of all F σ ideals is between \({\mathfrak{h}}\) and \({\mathfrak{b}}\), consistently different from both.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balcar B., Pelant J., Simon P.: The space of ultrafilters on N covered by nowhere dense sets. Fund. Math. 110(1), 11–24 (1980)
Bartoszyński T., Judah H.: Set theory: on the structure of the real line. A. K. Peters, Wellesley (1995)
Bartoszyński T., Shelah S.: Intersection of \({ < 2^{\aleph_{0}}}\) ultrafilters may have measure zero. Arch. Math. Log. 31(4), 221–226 (1992)
Blass, A.: Combinatorial cardinal characteristics of the continnum. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory, vol. 1. pp. 395–489. Springer, New York (2010)
Brendle J., Hruš ák M.: Countable fréchet boolean groups: an independence result. J. Symb. Log. 74(3), 1061–1068 (2009)
Dow A.: Tree π-bases for β N−N in various models. Topol. Appl. 33, 3–19 (1989)
Hernández-Hernández F., Hruš ák M.: Cardinal invariants of analytic P-ideals. Can. J. Math. 59(3), 575–595 (2007)
Hrušák, M.: Combinatorics of filters and ideals. In Set theory and its applications (Boise, ID, 1995–2010), vol. 533 of Contemp. Math., Am. Math. Soc. 29–69 (2011)
Hrušák M., Meza-Alcántara D., Minami H.: Pair-splitting, pair-reaping and cardinal invariants of F σ ideals. J. Symb. Log. 75(2), 661–677 (2010)
Hrušák, M., Rojas, D., Zapletal, J.: Cofinalities of borel ideals. Preprint
Kunen , K. : Set Theory. An Introduction to Independence Proofs. North Holland, Amsterdam (1980)
Kunen, K.: Random and Cohen Reals, Handbook of Set–Theoretic Topology (Kunen, K., Vaughan, J.E. eds.), North–Holland, Amsterdam (1984), 887–911
Mathias A.R.D.: Happy families. Ann. Math. Log. 12(1), 59–111 (1977)
Mazur K.: F σ-ideals and ω1ω *1 -gaps in the boolean algebras\({\mathcal{P}(\omega)/I}\). Fund. Math. 138(2), 103–111 (1991)
Pawlikowski, J.: Laver’s forcing and outer measure. In Set theory (Boise, ID 19921994) vol. 192 of Contemp. Math., pp. 71–76. Am. Math. Soc., Providence, RI 1996
Plewik S.: Intersection and unions of ultrafilters without the Baire property. Bull. Pol. Acad. Sci. Math. 35(11–12), 805–808 (1987)
Plewik S.: Ideals of second category. Fund. Math. 138(1), 23–26 (1991)
Solecki S.: Analytic ideals and their application. Ann. Pure Appl. Logic 99(1–3), 51–72 (1999)
Talagrand M.: Compacts de fonctions mesurables et filtres non mesurables. Stud. Math. 67(1), 13–43 (1980)
Zapletal J.: Forcing Idealized, Volume 174 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors gratefully acknowledge support from PAPIIT grant IN102311 and CONACyT grant 80355.
The third author is also supported by CONACyT, scholarship 209499 and the fourth author by CONACyT, scholarship 189774.
Rights and permissions
About this article
Cite this article
Hrušák, M., Martínez-Ranero, C.A., Ramos-García, U.A. et al. Intersection numbers of families of ideals. Arch. Math. Logic 52, 403–417 (2013). https://doi.org/10.1007/s00153-012-0321-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00153-012-0321-8