Date: Feb 22, 2013 6:05 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 222 Back to the roots

On 22 Feb., 02:08, William Hughes <wpihug...@gmail.com> wrote:
> So
>
>      A) For every natural number n, P(n) is true.
>
>         implies
>
>      B)  For any n: There does not exist a natural number
>          between 1 and n such that P(n) is false
>
> However, we cannot conclude
>
>      B')  There does not exist a natural number
>           m such that P(m) is false


In potential infinity there is no actually infinite set, i.e., there
is no completed infinity. That means there are only finite sequences.
However we cannot find a last element, although we can conclude that
it exists.

Compare my example:

A B
--> 1 -->{ }
--> 2,1 -->{ }
--> 2 -->1
--> 3, 2 -->1
--> 3 -->1, 2
--> 4, 3 -->1, 2
--> 4 -->1, 2, 3
...
--> n -->1, 2, 3, ..., n-1
--> n+1, n -->1, 2, 3, ..., n-1
--> n+1 -->1, 2, 3, ..., n-1, n
...

There is no last element in B. Every x from 1 to every n you desire
can be in B. Nevertheless there is never an empty A.

That is all we can know about infinity. (Not my fault.)

Regards, WM