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Topic: Matheology § 246
Replies: 246   Last Post: Apr 26, 2013 2:37 AM

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 mueckenh@rz.fh-augsburg.de Posts: 18,076 Registered: 1/29/05
Re: Matheology § 246
Posted: Apr 12, 2013 5:27 AM

On 12 Apr., 10:41, Virgil <vir...@ligriv.com> wrote:
> In article
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > Matheology § 246
>
> > Cantor's list contains real numbers r as binary or decimal fractions.
> > Real numbers, however, are /limits/ of binary or decimal fractions.

>
> Real numbers are all certainly expressable as limits of sequences, but
> so are rationals and integers and naturals.

No objection.

> > For every terminating fraction of r, Cantor obtains a difference
> > between r and the due terminating fraction of the anti-diagonal d:
> > r_nn =/= d_n. He concludes that this remains true for the limits of
> > the list numbers r and d by using the argument: different sequences
> > have different limits. But it is well known that this argument is not
> > admissible in proofs because it is false.

>
> The Cantor antidiagonal argument does not use the limit properties of
> sequences at all. In fact his binary sequences of members of {m,w} are
> not assumed to have any numerical or topological properties at all,

They are assumed to have infinitely many bits. But an infinite
sequence of bits is not converging (the factors 2^-n are missing).
Most of them cannot be written by finite expressions. And they cannot
be written as infinite expressions.

> much
> lesimits. Each of the Cantor sequences  is, in fact, merely a function
> from |N to {m,w} with no other properties assumed or allowed, except
> that two such functions are different if the is some n in |N at which
> they have different values.

Most of them are undefined and undefinable. But in order to remedy
that problem we use rationals in binary representation. Cantor's
argument, if valid, has to stand that example.

But it does not.

Regards, WM

Date Subject Author
4/12/13 mueckenh@rz.fh-augsburg.de
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