> In particular, the reals mean you accept measure theory, which in one of its key results says you can build two spheres of radius x from one sphere of radius

> In particular, the reals mean you accept measure theory, which in one of its key results says you can build two spheres of radius x from one sphere of radius x, and no holes. The computables don't go there. You cannot accept or reject measure theory. There are no axioms in measure theory; there are just the definition of a measure and theorems about measures. You're actually talking about the Axiom of Choice. If AC is true, then you can construct subsets of the reals that aren't measurable, and if AC is not true, then you can have a model of set theory where all subsets of the reals actually are measurable.