Re: Cantor's diagonal proof wrong?

From: W. Mueckenheim (mueckenh_at_rz.fh-augsburg.de)
Date: 11/30/04 

Date: 30 Nov 2004 07:33:31 -0800

Horst Kraemer <h-kraemer@lycos.de> wrote in message news:<qgslq0l4tg597qohvrvrehp42jc2qkf37f@4ax.com>...
> mueckenh@rz.fh-augsburg.de (W. Mueckenheim) wrote:
>
> > Sorry, but you are mistaken! Cantor does not (only) intend to show
> > that the exchanged diagonal differs from the "indicated limit", i. e.
> > the former value of the diagonal, but he has to show that the
> > exchanged diagonal is in fact different from any real number given in
> > his original Cantor-list. And that proof is valid only, if the
> > exchanged diagonal differs from any real number of the original
> > Cantor-list by at least one digit. Any consideration of an "indicated
> > limit" is completely irrelevant in this respect.
>
> And it does because by definition the "exchanged diagonal" differs
> from *every* real in the list by at least one position.
>
> It differs from real[k] in position k, for every k. because it is
> constructed like this. And as there are no reals in the list but those
> who have a number k, it differs from *every* real in the list.

Your arguing is true only for finite sequences D_k of the diagonal.
The number of digits of the complete diagonal D is infinite (aleph_0),
hence complete induction and your proof (or construction) fail.

If you don't accept this reasoning (I wouldn't do either), then you
must accept the reasoning below.
>
> If you are still assuming that there might be a real in the list from
> which it does not differ you are denying that
>
> It differs from every single real in the list
>
> is logically equivalent to
>
> There is no real in the list from which it does not differ
>
> is logically equivalent to
>
> There is no real in the list which is equal to it
>
> Which of the two assumed logical equivalences do you not accept ?

All three forms are equipollent, but wrong in case of the following
list:

0.0
0.1
0.11
0.111

The diagonal squence D_n constructed by a'_nn = 1 + a_nn is always
contained in line Z(n+1). This is correct for every finite n by
construction of the matrix.

This fact does never change. Therefore it is true for the whole
diagonal D, unless it contains elements with non-natural numbers as
indices.

You see, both statements "differs by definition" and "doesn't differ
by definition" can be applied with exactly the same justification.

There is not unique resolution.

Regards, WM

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