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 $b=d$? 

No choice is required. It suffices to exhibit an uncountable collection of real numbers which is linearly independent over $\mathbb{Q}$, and explicit choice-free examples are given in the answers to this MO question. 
