Diamond principles and Tukey-top ultrafilters on a countable set
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866917629407526912 |
|---|---|
| author | Benhamou, Tom Wu, Fanxin |
| author_facet | Benhamou, Tom Wu, Fanxin |
| contents | We provide two types of guessing principles for ultrafilter ($\diamondsuit^{-}_λ(U), \ \diamondsuit^p_λ(U)$) on $ω$ which form subclasses of Tukey-top ultrafilters, and construct such ultrafilters in $ZFC$. These constructions are essentially different from Isbell's construction \cite{Isbell65} of Tukey-top ultrafilters. We prove using the Borel-Cantelli Lemma that full guessing is not possible and rule out several stronger guessing principles e.g. we prove that no Dodd-sound ultrafilters exist on $ω$. We then apply these guessing principles to force a $q$-point which is Tukey-top (answering a question from \cite{Benhanou/Dobrinen23}), and prove that the class of ultrafilters which satisfy $\neg\diamondsuit^{-}_λ$ is closed under Fubini sum. Finally, we show that $\diamondsuit^{-}_λ$ and $\diamondsuit^p_λ$ can be separated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02379 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Diamond principles and Tukey-top ultrafilters on a countable set Benhamou, Tom Wu, Fanxin Logic General Topology We provide two types of guessing principles for ultrafilter ($\diamondsuit^{-}_λ(U), \ \diamondsuit^p_λ(U)$) on $ω$ which form subclasses of Tukey-top ultrafilters, and construct such ultrafilters in $ZFC$. These constructions are essentially different from Isbell's construction \cite{Isbell65} of Tukey-top ultrafilters. We prove using the Borel-Cantelli Lemma that full guessing is not possible and rule out several stronger guessing principles e.g. we prove that no Dodd-sound ultrafilters exist on $ω$. We then apply these guessing principles to force a $q$-point which is Tukey-top (answering a question from \cite{Benhanou/Dobrinen23}), and prove that the class of ultrafilters which satisfy $\neg\diamondsuit^{-}_λ$ is closed under Fubini sum. Finally, we show that $\diamondsuit^{-}_λ$ and $\diamondsuit^p_λ$ can be separated. |
| title | Diamond principles and Tukey-top ultrafilters on a countable set |
| topic | Logic General Topology |
| url | https://arxiv.org/abs/2404.02379 |