For the purposes of this discussion, let a Vitali Set be any subset $V\subseteq{}[0,1)$ such that for $V_q:=\{x+q\;|\;x<1-q,\;x\in{}V\}\cup\{x+q-1\;|\;x\geq{}1-q,\;x\in{}V\}$ there is a countable subset $I\subset[0,1)$ such that

- $[0,1)=\bigcup{}_{q\in{}I}V_q$
- For $r,q\in{}I$ distinct, $V_r\cap{}V_q=\emptyset$

Can such a $V$ be constructed without AC?