Date: Feb 23, 2013 5:39 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 222 Back to the roots

On 23 Feb., 00:03, William Hughes <wpihug...@gmail.com> wrote:
> Does
>
> For every natural number n, P(n)
> is true.
>
> imply
>
> There is no natural number m such
> that P(m) is false.


No. Only for all natural numbers n, P(n) is true would imply this.
If there is no P(n) for all natural numbers, then we are left with
every, but that holds only for all we can name. However, there are
always infinitely many we cannot name. Among them there can be natural
numbers that we cannot cannot identify, nevertheless we can prove that
they exist.

Remember my example with sets A and B. We cannot identify a last
element of the ordered set B and we cannot identify the element of A.
Nevertheless we can prove that there must be a last element, since A
is never empty by definition of A.

Regards, WM