[btilly]

Yes, you can write down Cantor's diagonal proof. But if you're careful about it, you find that the diagonalized thing is not a real number. For example a program to list programs that can be proven (by some set of axioms) to create Cauchy sequences can be used to create a Cauchy sequence, but you can't prove that that sequence is a Cauchy sequence without running into Gödel's incompleteness theorem.