HL ideals and Sacks indestructible ultrafilters

0576354 - MÚ 2025 RIV NL eng J - Journal Article
Chodounský, David - Guzmán, O. - Hrušák, M.
HL ideals and Sacks indestructible ultrafilters.
Annals of Pure and Applied Logic. Roč. 175, č. 1 (2024), č. článku 103326. ISSN 0168-0072. E-ISSN 1873-2461
Institutional support: RVO:67985840
Keywords : Halpern-Läuchli * ideal * Sacks forcing * ultrafilter
OECD category: Pure mathematics
Impact factor: 0.8, year: 2022
Method of publishing: Limited access
https://doi.org/10.1016/j.apal.2023.103326

We study ultrafilters on countable sets and reaping families which are indestructible by Sacks forcing. We deal with the combinatorial characterization of such families and we prove that every reaping family of size smaller than the continuum is Sacks indestructible. We prove that complements of many definable ideals are Sacks reaping indestructible, with one notable exception, the complement of the ideal Z of sets of asymptotic density zero. We investigate the existence of Sacks indestructible ultrafilters and prove that every Sacks indestructible ultrafilter is a Z-ultrafilter.
Permanent Link: https://hdl.handle.net/11104/0345923