On 19 Jun., 15:06, stevendaryl3...@yahoo.com (Daryl McCullough( wrote: > 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. > > 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.
All finite definitions of numbers map on infinite sequences of digits. But infinite sequences of digits do not map on finite definitions.
It is not possible to define a number by an infinite sequence of digits without having the finite definition.
Cantor's list consists of infinite sequences of digits. They are not defined and they do not define anything.
Only finite definitions define numbers. But an infinite list of finite definitions cannot be diagonalized.