Date: Apr 5, 2013 4:57 PM
Author: Virgil
Subject: Re: Matheology � 224

In article 
WM <> wrote:

> On 5 Apr., 12:08, William Hughes <> wrote:
> > On Apr 5, 11:09 am, WM <> wrote:
> >

> > > On 4 Apr., 23:08, William Hughes <> wrote:
> >
> > > > Nope.  Any single element can be removed.  This does not
> > > > mean the collection of all elements can be removed.

> >
> > > You conceded that any finite set of lines could be removed.  What is
> > > the set of lines that contains any finite set? Can it be finite? No.

> > correct
> > > So the set of lines that can be removed form an infinite set.
> >
> > More precisely.  There is an infinite set of lines D
> > such that any finite subset of D can be removed.

> What has to remain?

If one has any set, S, which is order isomorphic to the set of naturals
with their natural well-ordering, one can form the family, F, of FISs of
that set (finite initial segments).

Then any infinite subset of F will union to give the original S but no
finite subset of F will union to give back S.

That WM seems incapable of comprehending this simple truth marks his as
mathematically incompetent.
> >
> > This does not imply that D can be removed.

> > It does however imply that there is no single element
> > of D that cannot be removed. That this does not
> > imply that D can be removed is a result that
> > you do not like, but it is not a contradiction.

> It is simple mathological blathering to insist that |N contains only
> numbers that can be removed from |N but that not all natural numbers
> can be removed from |N.

Nonsense, Removing any member of |N from |N leaves a proper subset of |N.
However, removing FISONs from the set of all FISONs of |N may well leave
enough (infinitely many) to have their union equal |N.

Being unable to understand this seems to be WM's personal pons asinorum.
> It is a contradiction with mathematics, namely with the fact that
> every non-empty set of natural numbers has a smallest element.

Another wild false claim by WM made, as usual, without proof.