Date: Feb 23, 2013 5:45 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 222 Back to the roots

On 22 Feb., 23:39, Virgil <vir...@ligriv.com> wrote:
> In article
>
>
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 22 Feb., 22:29, Virgil <vir...@ligriv.com> wrote:
> > > In article

>
> > >  WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > 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.
>
> > > There is no such thing as an "A" but only an infinite sequence of
> > > differing A's, indexable by the infinite set of natural numbers,

>
> > In any case there is never an A = { }.
> > Therefore similarly there is never a B = |N.

>
> There is never an A or a B which is a subset of |N either, though their
> members are subsets of |N.

You are in error.
The set A_1 = {1} is a subset of |N.
The set B_1 = { } is a subset of |N

Regards, WM