The axiom of choice is not really about assigning names to objects using finite strings of symbols. In this example, you take R, remove countably infinitely many elements, and end up with a set with uncountably many elements. From a set theoretic perspective, it is not any more difficult or unintuitive to select an element from this set using AoC than it would be for R. When we use AoC, we are not really assigning a name to anything or doing any computation. We merely claim that a function exists which returns some element from the set. AoC doesn't compute or describe this function.