Date: Apr 5, 2013 4:40 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 224

On 5 Apr., 21:03, William Hughes <wpihug...@gmail.com> wrote:
> On Apr 5, 6:04 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>

> > On 5 Apr., 12:08, William Hughes <wpihug...@gmail.com> wrote:
>
> <snip>
>

> > > There is an infinite set of lines D
> > > such that any finite subset of D can be removed.

>
> > What has to remain?
>
> This depends on the finite subset removed.
> If the finite set removed is E then
> D\E has to remain.  Note that whatever
> subset E is chosen the number of lines
> in D\E is infinite


How do you call a set E the number of elements exceeds any given
natural number? (You do not claim that we can only remove a set E with
less than a given natural number, do you?)

>  (but of course we
> do not know which lines are in D\E).


How do we call a set when we cannot biject it with a FIS on |N?


> However, D cannot be removed without
> changing the union of the remaining lines.


That is correct, if D is not more than the union of all finite lines.

But if so, then every Cantor list that contains all rational numbers
has the following property:

For every n in |N: There are infinitely many lines that have the same
finite initials sequence d_1, d_2, d_3, ..., d_n of digits as the anti-
diagonal.

Only if the diagonal is more than every FIS, i.e., the list is more
than every finite lines, then this proof could be objected. For all n
in |N in is valid.

So you are caught in a circulus vitiosus: Either actual infity D is
more than all its FISs, then all can be removed without changing which
is obviously nonsense, or D is not more, then there is a proof against
Cantor's theorem which is as valid as Cantor's proof.

Regards, WM