Re: Cantor's diagonal proof wrong?

From: W. Mueckenheim (mueckenh_at_rz.fh-augsburg.de)
Date: 11/28/04 

Date: 28 Nov 2004 04:19:55 -0800

David Kastrup <dak@gnu.org> wrote in message news:<x5act3gdfz.fsf@lola.goethe.zz>...

> > However, how, then, can Cantor change all the digits of this
> > "limit"?
>
> He changes the digits of the sequence, resulting in a different value
> as the indicated limit.

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 why can't we consider my proof in the limit?
>
> You are again trying to befuddle people by word games. "considering a
> proof in the limit" is a completely meaningless and nonsensical
> expression.
>
> If we try to turn your word games into something making even remotely
> sense, the question might have been "Why does my proof that is valid
> for every element of a sequence say nothing about the limit of the
> sequence?". And the answer is, of course, that the limit is not a
> member of the sequence.

As I tried to express above already, Cantor's proof has nothing to do
with any limit, but only with one element a_nn for each number Z(n)
given in line number n.

  Even if we wanted to consider the limit as
> some weird element with index omega or whatever, for whatever crazy
> reason, your proof is an induction proof, and induction makes a
> statement only for all elements of N, and such a hypothetical omega
> could not be a member of N.

Of course, but if my proof should break down, it would do so only for
some miraculous omega. It remains valid for every natural n and every
number Z(n) in line number n.
>
> So whatever one likes to call the limit of the sequence of truncated
> decimals of D, your proof does not extend to it.

I didn't intend that, because Cantor didn't too. My intention is to
contradict Cantor, and not some miracle created by others.
>
> >> That the value specified by the diagonal (which is basically the
> >> limit of the corresponding series) can't be completely specified if
> >> you stop at any digit.
> >
> > So don't stop, but consider the limit of my proof.
>
> Proofs have no "limit". This is nonsensical. They have validity.
> Your proof is valid for all numbers corresponding to a truncated
> diagonal. The number corresponding to the entire diagonal is not
> covered by it.

Which number is that? Please give a plain answer.
Has Cantor to be satisfied with a truncated diagonal too? Or how can
he exchange further elements not covered by my numbers.
 
Regards, WM

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