> > It is proof that there is no countable set of all real numbers, since > any alleged such set is provably and constructably incomplete. > > Similarly, it is proof that there is no countable set of all > constructable numbers, since any alleged such set is provably and > constructably incomplete.
I hate to disagree with you, because we are on much the same "side", but this is not correct. Cantor's proof shows that you cannot form a list of all Reals. This is not the same as the Reals being uncountable.
You can use Cantor's diagonal construction to similarly prove that you cannot form a list of all computable numbers. However the computable numbers are in fact countable. You can't simply equate the two concepts; they are not exactly the same thing.