Date: Apr 17, 2013 11:42 PM
Author: Virgil
Subject: Re: Matheology � 222 Back to the roots

In article 
<2d6fce68-b735-4baf-8f75-d909189e0599@h9g2000vbk.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 15 Mrz., 13:56, William Hughes <wpihug...@gmail.com> wrote:
> > On Mar 14, 10:31 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> >
> >
> >
> >

> > > On 14 Mrz., 08:39, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > On Mar 13, 11:05 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> >
> > > > > On 13 Mrz., 22:41, William Hughes <wpihug...@gmail.com> wrote:
> >
> > > > > > Let J be a set of the lines of L with no
> > > > > > findable last line.  At least two lines
> > > > > > belong to J.  Are any lines of J necessary?

> >
> > > > > Remove all lines.
> > > > > Can any numbers remain in the list? No.
> > > > > Therefore at least one line must remain in the list.

> >
> > > > > We do not know which it is, but it is more than no line.
> > > > > In other words, it is necessary, that one line remains.

> >
> > > > However, it is not necessary that any one particular
> > > > line remain.  So while it is necessary that the set
> > > > J contain one line, there is no particular line l that is
> > > > necessary.

> >
> > > Correct. But I have not claimed that there are particular lines.
> >
> > Then it is a mistake to call particular line necessary
> > e.g. to say "There is a necessary line".-

>
> I don't say that there is a necessary line. It is necessary, that
> there remains a line or two. In detail:
>
> In potential infinity it is necessary that at least one line has to
> remain undeleted in order to contain all natural numbers that are in
> the list.
>
> In actual infinity it is necessary that at least two lines have to
> remain undeleted in order to contain all natural numbers of |N.
>
> No special line is necessary. But we know that two or more lines can
> never do a better job than one. So whatever lines may remain, the
> assertion is falsified.

At least two does not mean that two are sufficient.

It has been proven that infinitely may lines/FISONs are both neccesssary
and sufficient to contain all members of |N, but that no finite set of
FISONs is sufficient to contain all members of |N.
--