>Clearly there are countable sets that cannot be listed. > >Cantor proved that the Reals cannot be listed. His diagonal proof does *not* >show they are uncountable, any more than doing exactly the same proof with >computable Reals "proves" they are uncountable.
That's completely wrong. Cantor's proof shows that there is no bijection between N and R, which by definition means that the reals are uncountable.
It does *not* show that the computable reals are uncountable, because the antidiagonal is not necessarily a computable real.