1 results
$\mathscr {I}$-ULTRAFILTERS IN THE RATIONAL PERFECT SET MODEL
- Part of
-
- Journal:
- The Journal of Symbolic Logic / Volume 89 / Issue 1 / March 2024
- Published online by Cambridge University Press:
- 12 December 2022, pp. 175-194
- Print publication:
- March 2024
-
- Article
-
- You have access
- HTML
- Export citation
-
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.
