|
|
Re: Matheology § 201
Posted:
Jan 27, 2013 12:46 PM
|
|
On 27 Jan., 18:32, William Hughes <wpihug...@gmail.com> wrote:
> On Jan 27, 6:05 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
>
> <snip>
>
> >..the diagonal
> > cannot differ from all lines
> > (it differs from every line, though).
>
> The fact that the diagonal differs from every line is
> enough to show (induction) that the diagonal is not
> equal to any line in the list.
No. Induction shows that induction never proves something for *all*
naturals (simply because in PA there are not *all* naturals. Otherwise
AxInf would not be required in ZF.) Induction shows for *every* line n
that the diagonal is not in the first n lines. But induction also
shows that upon every line n there are infinitely many further lines
following. (Here potential infinty is meant, i.e., the number is not
larger than every finite number, but is not limited by any finite
number.)
You should try to distinguish these important facts.
Regards, WM
|
|
