In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> > > when replacing 0 by 1 has an anti-diagonal, the FIS of which are > > > always in the next line. So the anti-diagonal is not different from > > > all lines, unless it has an infinite sequence of 1's. But, as we just > > > saw, this is impossible. > > > > I see no significant difference between referring to a mathematical > > object by a formula and referring to it by 'writing it down' . > > But Cantor's argument is invalid, in this special case, unless it can > produce 0.111... with actually infinitely many 1's, i.e. more than > every finite number of 1's.
Consider an endless list of terminating decimals in which the nth decimal place of the nth decimal is a 5.
Or more generally any list of reals whose nth decimal digit is greater than 1.
Then, for ANY and EVERY such list, 1/9, with always a 1 in its nth decimal place, will be a legitimate anti-diagonal to such a list.
So once again, WM is proven totally and foolishly wrong about what is or is not the case with Cantor diagonals. --