In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 21 Jun., 12:41, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > > Peter Webb says... > > > > >Cantor constructs the antidiagonal by taking the list and then computing a > > >missing Real based upon the decimal expansion of every Real on the list. > > > > That's right, so Cantor showed how to compute an antidiagonal > > using the list of reals as an input. That shows that the antidiagonal > > is computable *relative* to the list. It does not show that the > > antidiagonal is computable. "Computable relative to a list" and > > "computable" are two different things. > > > > This is a pretty simple concept, Peter. Mathematically, > > you have a function antidiagonal(L,n) which returns the > > nth digit of the antidiagonal, given the list L and n. > > That function is computable. But for the antidiagonal > > to be computable (not computable *relative* to L), you > > would have to be able to come up with a function > > > > antidiagonal(n) > > > > that, for *any* n, gives the nth digit of the antidiagonal. > > > > Two different concepts: computable relative to L, and computable. > > But none of them does help to save set theory.
WM is the only who seems to have thrown away set theory, but, despite WM's urgings, almost no one else does. > > The diagonal numbers computable relative to a list are countable. And > so are the diagonal numbers computable relative to a list that is > extended in the n-th step by the diagonal number computed in step n - > 1. > > Nevertheless, there is not list of these "relative to the list" > computable diagonal numbers. Therefore Cantor's proof shows the > uncountability of a countable set.
By at least some definitions of "computable number", the anti-diagonal of a list of computable numbers need not be itself computable, and if not, the set of computable numbers should be countable.
And I am not at all sure that for definitions of "computable number" which would require such an anti-diagonal to be computable that the set of computable numbers is provably countable.