Re: Cantor's diagonal proof wrong?

From: W. Mueckenheim (mueckenh_at_rz.fh-augsburg.de)
Date: 12/01/04 

Date: 30 Nov 2004 22:24:21 -0800

"Shmuel (Seymour J.) Metz" <spamtrap@library.lspace.org.invalid> wrote in message news:<41aa5d5a$14$fuzhry+tra$mr2ice@news.patriot.net>...
> In <fb701d3c.0411271334.cca9c93@posting.google.com>, on 11/27/2004
> at 01:34 PM, mueckenh@rz.fh-augsburg.de (W. Mueckenheim) said:
>
> >What does distinguish the "limit" from the diagonal?
>
> What distinguishes a banana from a townhouse? They're not remotely
> similar.
>
> The diagonal is a sequence of digits, not a number. In the case of the
> Cantor diagonal argument you're referring to, you take the sequence of
> digits as coefficients in a series and it is trivial to prove that the
> series converges.

Ok, the abbreviation "diagonal" should expand to "diagonal number".
>
> >Do we need different words here?
>
> We need words that are applicable, and we need to ensure that we have
> a common understanding of what they refer to. In particular, the term
> limit has a precise meaning.
>
> >However, how, then, can Cantor change all the digits of this
> >"limit"?
>
> He doesn't "change" anything. He defines a new number in terms of a
> sequence of representations of numbers.

He defines it by changing the digits a_nn of the diagonal number D to
a'_nn of the new number D', because he must make sure that in any case
a'_nn /= a_nn.

> >And why can't we consider my proof in the limit?
>
> What proof?

I have definined a Cantor-list, which always contains the diagonal
number D_n constructed up to line n in line Z(n+1) by construction. I
found this very same list also appearing in this thread:

0.000...
0.100...
0.11000...
0.111000...
...

Changing the diagonal elements 0 -> 1, we have D_n = Z(n+1).

We see that either of the two statements:
A) Cantors changed diagonal number differs from every real in a line
not A) Cantors diagonal number does not differ from every real in a
line
can be taken for granted. There is no logical priority in favour of A
or not A, as long as all lines are enumerated by natural, hence finite
numbers.

Regards, WM

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