Let $\mathbb{F}_2 = \{0,1\}$ be the field with two elements.
The naturals $\mathbb{N} = \{0,1,2,3,...\}$ (with vector addition given by binary bitwise XOR) form a vector space over $\mathbb{F}_2$, with a Hamel basis $\{2^n \mid n \in \mathbb{N}\}$. The dyadic rationals $\mathbb{N}[\tfrac{1}{2}]$, with vector addition defined the same way, also form a vector space over $\mathbb{F}_2$ with a Hamel basis $\{2^k \mid n \in \mathbb{Z}\}$.
Viewing $\mathbb{R}_{\geq 0}$ similarly as a vector space over $\mathbb{F}_2$, what does a Hamel basis look like? I cannot define its vector addition explicitly as this is an open problem. However, this isn't necessary; my question boils down to finding some $\mathcal{B} \subset \mathbb{R}_{\geq 0}$ such that:
- Every $r \in \mathbb{R}_{\geq 0}$ can be identified with a finite subset $X \subset \mathcal{B}$ such that $r = \sum_{x \in X}x,$
- This representation is unique; $\sum_{x \in X}x = \sum_{y \in Y}y \iff X=Y$.