It is easy to find a choice function on all finite subsets of $\mathbb R$, but without using the axiom of choice, not on all subsets. Is there an "explicit" choice function on the countable subsets of the real line? Can it be used to create paradoxical objects like, say, Vitali sets in the same way that a choice function on all subsets does?
Edit: Chris notes that the construction of the Vitali set only involves choosing from countable subsets, so there is no hope for such a function to be explicit. I revise my question, then: does the existence of a choice function on all countable subsets imply the existence of a global choice function? Is it consistent that there is a choice function on the countable subsets, but not on all subsets?