Well the issue is whether its possible in principle to define such a function for all possible sets. But if you can enumerate the elements of the set, such a fu

Well the issue is whether its possible in principle to define such a function for all possible sets. But if you can enumerate the elements of the set, such a function is simply to pick the first element. So the issue is how much information is necessary to guarantee that its possible in principle to construct such a function. It should be clear that enough information to enumerate each set is enough to create a decision function. The remaining question seems to be whether one can specify a countable set of countable sets without specifying enough info about each member set to enumerate the set's members. It seems trivial that a set with countable elements is enumerable, but I might be missing some subtlety.