Date: Mar 24, 2013 7:50 AM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 224

On 23 Mrz., 19:08, "Mike Terry"
<news.dead.person.sto...@darjeeling.plus.com> wrote:
> "David R Tribble" <da...@tribble.com> wrote in messagenews:85d0c23d-7dcd-4607-b22d-d2444df80433@googlegroups.com...
>

> > WM wrote:
> > >>... consider the list of finite initial segments of natural numbers
> > > 1
> > > 1, 2
> > > 1, 2, 3
> > > ...

>
> > > According to set theory it contains all aleph_0 natural numbers in its
> > > lines. But is does not contain a line containing all natural numbers.
> > > Therefore it must be claimed that more than one line is required to
> > > contain all natural numbers. This means at least two line are
> > > necessary.

>
> > That is correct. In fact, all Aleph_0 lines are required
> > (necessary sufficient) to contain all of the naturals.

>
> This is sufficient but not necessary.  (Aleph_0 lines are necessary and
> sufficient.)
>

This is a false claim, if induction is valid and if |N has more
elements than every finite line.
For aleph_0 lines, namely every finite line, my proof shows that they
are not necessary.

Regards, WM