Date: Jan 30, 2013 4:46 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 203

On 30 Jan., 10:31, William Hughes <wpihug...@gmail.com> wrote:

> For a potentially infinite list L, the
> antidiagonal of L is not a line of L.


Of course. Every subset L_1 to L_n can be proved to not contain the
anti-diagonal
>
> Does this imply
>
> There is no potentially infinite list
> of 0/1 sequences, L, with the property that
> any 0/1 sequence, s, is one of the lines
> of L.


Do you mean potentially infinite sequences?
Look, everything Cantor does, concerns only finite initial segments.
You could cut off the sequences behind the digonal digit.

The only thing not terminating, then could be the diagonal itself. But
then you would claim that the diagonal differs from every entry,
because it has more digits. In the original argument, the diagonal
differs at the same places that also exist in the entries. Therefore
the argument with the diagonal "being longer" is wrong.

So in fact, Cantor shows that the countable set of all terminating
decimals is uncountable. Of course this proof is wrong, as using a
list that contains all terminating decimals shows. This argument only
shows that "countability" as a property of actual infinite sets is
nonsense.

Regards, WM