Monotonicity of the ultrafilter number function
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911277614366720 |
|---|---|
| author | Usuba, Toshimichi |
| author_facet | Usuba, Toshimichi |
| contents | We investigate whether the ultrafilter number function $κ\mapsto \mathfrak{u}(κ)$ on the cardinals is monotone, that is, whether $\mathfrak{u}(λ) \le \mathfrak{u}(κ)$ holds for all cardinals $λ< κ$ or not. We show that monotonicity can fail, but the failure has large cardinal strength. On the other hand, we prove that there are many restrictions of the failure of monotonicity. For instance, if $κ$ is a singular cardinal with countable cofinality or a strong limit singular cardinal, then $\mathfrak{u}(κ) \le \mathfrak{u}(κ^+)$ holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14988 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monotonicity of the ultrafilter number function Usuba, Toshimichi Logic 03E10, 03E35, 03E55 We investigate whether the ultrafilter number function $κ\mapsto \mathfrak{u}(κ)$ on the cardinals is monotone, that is, whether $\mathfrak{u}(λ) \le \mathfrak{u}(κ)$ holds for all cardinals $λ< κ$ or not. We show that monotonicity can fail, but the failure has large cardinal strength. On the other hand, we prove that there are many restrictions of the failure of monotonicity. For instance, if $κ$ is a singular cardinal with countable cofinality or a strong limit singular cardinal, then $\mathfrak{u}(κ) \le \mathfrak{u}(κ^+)$ holds. |
| title | Monotonicity of the ultrafilter number function |
| topic | Logic 03E10, 03E35, 03E55 |
| url | https://arxiv.org/abs/2501.14988 |