Date: Feb 23, 2013 4:43 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 222 Back to the roots

On 23 Feb., 22:15, Virgil <vir...@ligriv.com> wrote:
> In article
> <62781b70-dff9-4093-85d0-ff6e5bfcb...@u20g2000yqj.googlegroups.com>,
>
>
>
>
>
>  WM <mueck...@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.


A_1 is A in the first step.
>
> > The set B_1 = { } is a subset of |N
>
> But B_1 is merely a term of sequence B and not equal to B.


B_1 is B in the first setp.

Regards, WM