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Characterizing existence of certain ultrafilters
- Source :
- Ann. Pure Appl. Logic 173 (2022), no. 9, Paper No. 103157
- Publication Year :
- 2023
-
Abstract
- Following Baumgartner [J. Symb. Log. 60 (1995), no. 2], for an ideal $\mathcal{I}$ on $\omega$, we say that an ultrafilter $\mathcal{U}$ on $\omega$ is an $\mathcal{I}$-ultrafilter if for every function $f:\omega\to\omega$ there is $A\in \mathcal{U}$ with $f[A]\in \mathcal{I}$. If there is an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter, then $\mathcal{I}$ is not below $\mathcal{J}$ in the Kat\v{e}tov order $\leq_{K}$ (i.e. for every function $f:\omega\to\omega$ there is $A\in \mathcal{I}$ with $f^{-1}[A]\notin \mathcal{J}$). On the other hand, in general $\mathcal{I}\not\leq_{K}\mathcal{J}$ does not imply that existence of an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter is consistent. We provide some sufficient conditions on ideals to obtain the equivalence: $\mathcal{I}\not\leq_{K}\mathcal{J}$ if and only if it is consistent that there exists an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter. In some cases when the Kat\v{e}tov order is not enough for the above equivalence, we provide other conditions for which a similar equivalence holds. We are mainly interested in the cases when the family of all $\mathcal{I}$-ultrafilters or $\mathcal{J}$-ultrafilters coincides with some known family of ultrafilters: P-points, Q-points or selective ultrafilters (a.k.a. Ramsey ultrafilters). In particular, our results provide a characterization of Borel ideals $\mathcal{I}$ which can be used to characterize P-points as $\mathcal{I}$-ultrafilters. Moreover, we introduce a cardinal invariant which is used to obtain a sufficient condition for the existence of an $\mathcal{I}$-ultrafilter which is not a $\mathcal{I}$-ultrafilter. Finally, we prove some new results concerning existence of certain ultrafilters under various set-theoretic assumptions.
- Subjects :
- Mathematics - Logic
03E17 (Primary), 03E05, 03E35, 03E50, 03E75 (Secondary)
Subjects
Details
- Database :
- arXiv
- Journal :
- Ann. Pure Appl. Logic 173 (2022), no. 9, Paper No. 103157
- Publication Type :
- Report
- Accession number :
- edsarx.2308.12594
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1016/j.apal.2022.103157