Date: Feb 22, 2013 6:12 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 222 Back to the roots

On 21 Feb., 21:51, Virgil <vir...@ligriv.com> wrote:

> > Or consider the union of natural numbers in a set B while there
> > remains always one number in the intermediate reservoir A.

>
> >      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
> > ...

>
> > One would think that never all naturals can be collected in B, since a
> > number n can leave A not before n+1 has arrived.

>
> > Of course this shows that ZF with its set of all natural numbers is
> > contradicted.

>
> WM's A and B are not sets but sequences of sets, so if WM wants to
> consider a limit to any such sequences, he must first define what he
> means by such a limit, as there is no universal definition for "the"
>  limit of a sequence of sets.


By definition of A we know it is never empty. That implies that B
never contains all natural numbers. B always has a last element, but
we cannot know it, because if we say n, then n+1 is as well in B.

That is the property of infinity. I am not responsible for that
behaviour, I only recall what our ancestors knew.

Regards, WM