Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.01739 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Under $\mathrm{ZF}$, we show that the statement that every subset of every $\mathbb{R}$-vector space has a maximal convex subset is equivalent to the Axiom of Choice. We also study the strength of the same statement restricted to some specific $\mathbb{R}$-vector spaces. In particular, we show that the statement for $\mathbb{R}^2$ is equivalent to the Axiom of Countable Choice for reals, whereas the statement for $\mathbb{R}^3$ is equivalent to the Axiom of Uniformization. We discuss the statement for some spaces of higher dimensions as well.