>I agree that computable reals are countable. But I do not agree this means >they can be listed. In fact, I can easily prove they are not. If you give me >a purported list of all computable Reals I can use a diagonal argument to >form a computable Real not on the list.
You can use a diagonal argument to form a *real* that is not on the list. For that real to be *computable*, you need to show that you can compute that real *without* using the list.
If the list were a computable list, then you could reconstruct it yourself, so the antidiagonal would be computable. If the list is not computable, then neither is the antidiagonal.