Re: Cantor's diagonal proof wrong?

From: Horst Kraemer (h-kraemer_at_lycos.de)
Date: 12/01/04 

Date: Wed, 01 Dec 2004 09:47:34 +0100

mueckenh@rz.fh-augsburg.de (W. Mueckenheim) wrote:

> "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.

Why is

I) D_k =/= 0 for every k

II) for every Z_n:
           not ( Z_n,k =/= 0 for every k )

or, denoting E the property of a sequence to have *all* of its
elements =/= 0

I) D has the property E

II) no Z_n has the property E

*not* a logical priority for ( D differs from every Z_n ) ? Just
*because* all lines are enumerated by natural numbers from which
follows that there is no other candidate for equality than *numbered*
lines, and because a sequence can only be equal to the sequence D if
it *has* got the property E. 

-- 
Horst

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