"Daryl McCullough(" <firstname.lastname@example.org> wrote in message news:email@example.com... > Peter Webb says... > >>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. >
That is crap. Cantor uses the list. In fact, he uses the value of the nth digit of the nth term. The list is provided in advance. If the list were not provided in advance, you wouldn't be able to prove its missing something.
> 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.
A list of all computable Reals is not computable. A list of all Reals is not computable. The latter was proved by Cantor. It does not automatically prove that the Reals are uncountable, any more the the same proof applied to computable Reals proves they are uncountable.