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