choose $r(i)$ such that it is irrational and from $[0,1]$.
$r_1 - r_2 = q\in\mathbb{Q}$ implies its in an equivalence class.
Seems the equivalence class for $r_1$ has countably infinite members.
choose $r_3$ such that $r_1 < r_3 < r_2$ and $r_3 - r_1$ does not equal some $q\in \mathbb{Q}$. The equivalence class for $r_3$ then seems to have countably infinite irrational numbers as members.
This continues until finally exhausting all $r$ such that $r_1 < r < r_2$. Supposedly, there is now an uncountable number of equivalence classes, each with countably infinite members.
Is a Vitali set then a question of what is the measure of $r_2 - r_1$ for real numbers that are arbitrarily close to each other?