On 29 Nov., 18:35, William Hughes <wpihug...@hotmail.com> wrote: > On Nov 29, 8:05 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > WM has conceded that you can use induction > to show that every element of the list has > a final 1, and that there is a constructive > proof that the diagonal number does not have a > final 1. > > WM has a new argument. > > > Use induction to show that the diagonal number cannot have more digits > > than every entry of the list. > > This cannot be done. All you do is show that > every one of an infinite number of different > numbers, none of which is the diagonal number, > cannot have more digits than every entry of the list.
There is a simple proof by contradiction: Assume that the diagonal (in the example-list
0.0 0.1 0.11 0.111 ...)
has a digit that is not in an entry of the list. This would mean that the list has an end. That is wrong by definition. Therefore every seqeunce of 1's in the diagonal is in an entry of the list.