Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.12018 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911107360227328 |
|---|---|
| author | Hayashi, Yusuke |
| author_facet | Hayashi, Yusuke |
| contents | We study the generalized dominating number $\mathfrak{d}_μ$ at a singular cardinal $μ$ of cofinality $κ$. We show two lower bounds: in ZFC, $\mathrm{cf}([μ]^κ,\subseteq) \leq \mathfrak{d}_μ$, and under mild cardinal-arithmetic assumptions, $2^{<μ} \leq \mathfrak{d}_μ$. We also clarify when $\mathfrak{d}_μ$ can differ from $2^μ$: assuming GCH and $κ= \mathrm{cf}(μ) > ω$, a finite-support iteration of Cohen forcing of length $μ^{++}$ yields $\mathfrak{d}_μ < 2^μ$. On the other hand, for $κ= \mathrm{cf}(μ) = ω$, natural $μ$-cc posets force $\mathfrak{d}_μ = 2^μ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12018 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dominating numbers at singular cardinals Hayashi, Yusuke Logic We study the generalized dominating number $\mathfrak{d}_μ$ at a singular cardinal $μ$ of cofinality $κ$. We show two lower bounds: in ZFC, $\mathrm{cf}([μ]^κ,\subseteq) \leq \mathfrak{d}_μ$, and under mild cardinal-arithmetic assumptions, $2^{<μ} \leq \mathfrak{d}_μ$. We also clarify when $\mathfrak{d}_μ$ can differ from $2^μ$: assuming GCH and $κ= \mathrm{cf}(μ) > ω$, a finite-support iteration of Cohen forcing of length $μ^{++}$ yields $\mathfrak{d}_μ < 2^μ$. On the other hand, for $κ= \mathrm{cf}(μ) = ω$, natural $μ$-cc posets force $\mathfrak{d}_μ = 2^μ$. |
| title | Dominating numbers at singular cardinals |
| topic | Logic |
| url | https://arxiv.org/abs/2508.12018 |