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?



Choice function on the countable subsets of the reals
The standard construction of a [Vitali set](http://en.wikipedia.org/wiki/Vitali_set) only involves making choices from countable subsets of $\mathbb{R}$ (specifically, from sets of the form $(r+\mathbb{Q}) \cap [0,1]$). It is well known that ZF (assuming consistent) does not prove the existence of a non-measurable subset of $\mathbb{R}$, hence it doesn't prove the existence of a Vitali set, and thus doesn't prove your restricted choice principle.




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[Consequences of the Axiom of Choice](http://consequences.emich.edu/conseq.htm) is useful for this sort of thing. According to this, form 85 (every collection of countable sets has a choice function) is true while form 79 (every collection of sets of reals has a choice function) is false in model $\mathcal{M} 1$, which I believe is Cohen's original model of ~AC. Thus your countable choice principle for the reals does not imply full choice for the reals.


