Re: Cantor's diagonal proof wrong?

From: W. Mueckenheim (mueckenh_at_rz.fh-augsburg.de)
Date: 11/24/04 

Date: 24 Nov 2004 07:38:57 -0800

curt@kcwc.com (Curt Welch) wrote in message news:<20041122132635.076$bc@newsreader.com>...
> "*** T. Winter" <***.Winter@cwi.nl> wrote:
>
> > I think that is not what he meant. I think he meant that it is the
> > same as the first row with the *first* digit swapped. Similarly, the
> > third row is the same as the second, with the *second* digit swapped.
> > Using this algorithm we get:
> > 00000...
> > 10000...
> > 11000...
> > 11100...
> > and the diagonal is:
> > 11111...
> >
> > Now the interesting thing is that it looks like the diagonal is on
> > the list. But it obviously is not, for the same reason that the
> > infinite set of natural numbers does not contain an infinite number.
> > But CW is thinking that you have to construct initial parts of the
> > diagonal number sequentially, and that is where his statement
> > "every diagonal" comes from.
>
> Right. You have restored my faith in mainkind. :) You may not know why
> I'm thinking the way I think, but you at least show that you are smart
> enough and have enough of an open mind to understand something new and
> different.
>
> And you are right, the argument is exatly the same as my initial argument
> about infinite sized integers but the new form of the argument takes the
> definition of integers out of the picture. I thought taking integers out
> of the argument would be enought to clear up the confusion. But it wasn't.
> If you use the normal mathematical ideas about creation and existence, my
> argument still falls short even though integers have been removed from the
> language. So in the langauge of math, no version of my argument makes
> sense. Which is why I now say that what I've been talking about is
> actually an issue outside of math.

Given the sequence of terminating rationals already discussed above:
Can diagonalization, substituting 0 by 1, yield a number not contained
in the sequence? Is anybody able to show at which natural number this
argument fails. Or do we have to accept non-natural numbers
enumerating the list?

Regards, WM

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