Date: Feb 23, 2013 4:15 PM
Author: Virgil
Subject: Re: Matheology � 222 Back to the roots

In article 
<62781b70-dff9-4093-85d0-ff6e5bfcbbb8@u20g2000yqj.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 22 Feb., 23:39, Virgil <vir...@ligriv.com> wrote:
> > In article
> > <6cfcca98-d4e5-475d-a4bf-168639050...@n2g2000yqg.googlegroups.com>,
> >
> >
> >
> >
> >
> >  WM <mueck...@rz.fh-augsburg.de> wrote:

> > > On 22 Feb., 22:29, Virgil <vir...@ligriv.com> wrote:
> > > > In article
> > > > <c3c197e4-2161-4ecf-a84e-d479adb05...@k4g2000yqn.googlegroups.com>,

> >
> > > >  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.


But A_1 is merely a member of the sequence A, and is not A itself.

> The set B_1 = { } is a subset of |N

But B_1 is merely a term of sequence B and not equal to B.

SO it is WM who is in error by trying to make A stand for two different
things at once, and B stand for two different things at once.

WM's A and B can only stand for the whole of a sequence or for one term
in that sequence but not for both simultaneously, as he is trying to
make them do.
--