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

> <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.

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

Regards, WM