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GUZMÁN, OSVALDO
2023.
KATĚTOV ORDER ON MAD FAMILIES.
The Journal of Symbolic Logic,
p.
1.
Article contents
$\mathscr {I}$-ULTRAFILTERS IN THE RATIONAL PERFECT SET MODEL
Part of:
Set theory
Published online by Cambridge University Press: 12 December 2022
Abstract
We give a new characterization of the cardinal invariant 
$\mathfrak {d}$ as the minimal cardinality of a family 
$\mathcal {D}$ 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 
$\mathcal {D}$. On the other hand, we prove that in the Miller model, given any family 
$\mathcal {D}$ of analytic tall p-ideals such that 
$\vert \mathcal {D}\vert <\mathfrak {d}$, there is an ultrafilter 
$\mathcal {U}$ which is an 
$\mathscr {I}$-ultrafilter for all ideals 
$\mathscr {I}\in \mathcal {D}$ at the same time, yet 
$\mathcal {U}$ is not a rapid ultrafilter. As a corollary, we obtain that in the Miller model, given any analytic tall p-ideal 
$\mathscr {I}$, 
$\mathscr {I}$-ultrafilters are dense in the Rudin–Blass ordering, generalizing a theorem of Bartoszyński 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.
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