I-ultrafilters in the rational perfect set model

0585180 - MÚ 2025 RIV GB eng J - Journal Article
Cancino Manríquez, Jonathan
I-ultrafilters in the rational perfect set model.
Journal of Symbolic Logic. Roč. 89, č. 1 (2024), s. 175-194. ISSN 0022-4812. E-ISSN 1943-5886
Institutional support: RVO:67985840
Keywords : analytic p-ideal * I-ultrafilter * rapid ultrafilter
OECD category: Pure mathematics
Impact factor: 0.6, year: 2022
Method of publishing: Open access
https://doi.org/10.1017/jsl.2022.81

We give a new characterization of the cardinal invariant d as the minimal cardinality of a family D of tall summable ideals such that an ultrafilter is rapid if and only if it has non-empty intersection with all the ideals in the family D . On the other hand, we prove that in the Miller model, given any family D of analytic tall p-ideals such that |D| < d , there is an ultrafilter U which is an I -ultrafilter for all ideals I is an element of D at the same time, yet U is not a rapid ultrafilter. As a corollary, we obtain that in the Miller model, given any analytic tall p-ideal I , I -ultrafilters are dense in the Rudin-Blass ordering, generalizing a theorem of Bartoszynski and S. Shelah, who proved that in such model, Hausdorff ultrafilters are dense in the Rudin-Blass ordering. This theorem also shows some limitations about possible generalizations of a theorem of C. Laflamme and J. Zhu.
Permanent Link: https://hdl.handle.net/11104/0352925