Let us define an equivalence relation $\sim$ on $\mathbb{R}$ by saying that $x\sim y$ if $x-y\in \mathbb{Q}$? This equivalence relation partitions $\mathbb{R}$ into uncountably many equivalence classes. My question is, is it possible to construct a set which has exactly one element from each of these equivalence classes?
Can we define these elements explicitly? Or failing that, can we at least prove that there exists a definable subset of $\mathbb{R}$ which has this property? What about a Borel subset of $\mathbb{R}$?
Such a set is a Vitali set. It is non-measurable (and in particular not Borel). Solovay showed it is consistent with ZF (without Axiom of Choice) that all subsets of $\mathbb R$ are measurable. Therefore without some form of Axiom of Choice it is impossible to construct such a set and prove it has the desired property.