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.
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.