Does an uncountable Golomb ruler exist? Does there exist an uncountable set $G\subset \mathbb{R}$ such that, for $a,b,c,d \in G$, if $a-b=c-d$ then $a=c$ and $b=d$? 

I think so (assuming Choice). Let $G$ be a vector space basis of $\Bbb{R}$ over $\Bbb{Q}$. Then $G$ is, among other things, linearly independent over $\Bbb{Z}$ and uncountable.
