Abstract
Forcing notions of the type \(\mathcal{P}(\omega)/\mathcal{I}\) which do not add reals naturally add ultrafilters on ω. We investigate what classes of ultrafilters can be added in this way when \(\mathcal{I}\) is a definable ideal. In particular, we show that if \(\mathcal{I}\) is an F σ P-ideal the generic ultrafilter will be a P-point without rapid RK-predecessors which is not a strong P-point. This provides an answer to long standing open questions of Canjar and Laflamme.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Booth, D.: Ultrafilters on a countable set. Ann. Math. Log. 2(1), 1–24 (1970/1971)
Canjar, R.M.: Mathias forcing which does not add dominating reals. Proc. Am. Math. Soc. 104(4), 1239–1248 (1988)
Choquet, G.: Deux classes remarquables d’ultrafiltres sur N. Bull. Sci. Math. 92, 143–153 (1968)
Dow, A.: Tree π-bases for βℕ∖ℕ in various models. Topol. Appl. 33(1), 3–19 (1989)
Farah, I.: Analytic quotients: theory of liftings for quotients over analytic ideals on the integers. Mem. Am. Math. Soc. 148(702), xvi+177 (2000)
Hernández-Hernández, F., Hrušák, M.: Cardinal invariants of analytic P-ideals. Can. J. Math. 59(3), 575–595 (2007)
Hrušák, M., Minami, H.: Mathias-Prikry and Laver-Prikry type forcing (2010), preprint
Hrušák, M., Verner, J., Blass, A.: On strong P-points (2010), preprint
Hrušák, M., Zapletal, J., Rojas, D.: Cofinalities of Borel ideals (2010), preprint
Just, W., Krawczyk, A.: On certain Boolean algebras \(\mathcal{P}(\omega)/I\). Trans. Am. Math. Soc. 285(1), 411–429 (1984)
Katětov, M.: Products of filters. Comment. Math. Univ. Carol. 9, 173–189 (1968)
Laflamme, C.: Forcing with filters and complete combinatorics. Ann. Pure Appl. Log. 42(2), 125–163 (1989)
Mathias, A.R.D.: Happy families. Ann. Math. Log. 12(1), 59–111 (1977)
Mazur, K.: F σ -ideals and \(\omega_{1}\omega_{1}^{*}\)-gaps in the Boolean algebras \(\mathcal{P}(\omega)/I\). Fundam. Math. 138(2), 103–111 (1991)
Meza-Alcántara, D.: Ph.D. thesis, UNAM (2009)
Mokobodzki, G.: Ultrafiltres rapides sur N. Construction d’une densité relative de deux potentiels comparables, Séminaire de Théorie du Potentiel, dirigé par M. Brelot, G. Choquet et J. Deny: 1967/68, Exp. 12, Secrétariat mathématique, Paris (1969), p. 22
Rudin, W.: Homogeneity problems in the theory of Čech compactifications. Duke Math. J. 23, 409–419 (1956)
Solecki, S.: Analytic ideals. Bull. Symb. Log. 2(3), 339–348 (1996)
Solecki, S.: Analytic ideals and their applications. Ann. Pure Appl. Log. 99(1–3), 51–72 (1999)
Szymański, A., Xua, Z.H.: The behaviour of \(\omega^{2^{\ast} }\) under some consequences of Martin’s axiom. In: General Topology and Its Relations to Modern Analysis and Algebra, V (Prague, 1981). Sigma Ser. Pure Math., vol. 3, pp. 577–584. Heldermann, Berlin (1983)
Vojtáš, P.: On ω ∗ and absolutely divergent series. Topol. Proc. 19, 335–348 (1994)
Zapletal, J.: Preserving P-points in definable forcing. Fundam. Math. 204(2), 145–154 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of the first author was partially supported by PAPIIT grant IN101608 and CONACyT grant 80355.
The second author would like to acknowledge the support of GAČR 401/09/H007 Logické základy sémantiky.
Rights and permissions
About this article
Cite this article
Hrušák, M., Verner, J.L. Adding ultrafilters by definable quotients. Rend. Circ. Mat. Palermo 60, 445–454 (2011). https://doi.org/10.1007/s12215-011-0064-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12215-011-0064-0