But even if you have the set of all reals not describable with a finite amount of symbols, I can still imagine what happens e.g. if you multiply all its values by 2, or what the cartesian product of this set with another is like. At least, I can reason sbout that result in a way that's not worse than reasoning about the set itself. Using axiom of choice on it does not make things harder than the set itself already is
Banach-Tarski says:
Take a measurable set.
Split it up into 2 unmeasurable sets.
Recombine them into a set with twice the measure of the original.
OK, you can do that, but you've lost the ability to make any reasonable theorems about measure of measurable sets, unless you add qualifications of the form "this theorem does not apply if you introduce a set of uncountable sets" which. What's the benefit? To get a bunch of "proofs" for things that are only true because of the axiom, and don't model anything else we care about?
