Date: Feb 24, 2013 5:20 PM
Author: Virgil
Subject: Re: Matheology � 222 Back to the roots

In article 
<a07faa13-25cf-43de-8a4e-499f8e839339@l9g2000yqp.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 24 Feb., 15:56, William Hughes <wpihug...@gmail.com> wrote:
> > So, when WM says that a natural number m does not
> > exist, he may mean that you can prove it exists
> > but you cannot find it.
> >
> > Suppose that P is a predicate such that
> > for every natural number m, P(m) is true.

>
>
> Example: Every line of the list L
>
> 1
> 1, 2
> 1, 2, 3
> ...
>
> contains all its predecessors.
>
>

> >
> > Let x be a natural number such that
> > P(x) is false. According to WM you cannot
> > prove that x does not exist.  (WM
> > rejects the obvious proof by contradiction:
> >
> >      Assume a natural number, x, such that P(x)
> >      is false exists.
> >      call it k
> >      Then P(k) is both true and false.
> >      Contradiction,  Thus the original assumption
> >      is false and no natural number, x, such
> >      that P(x) is false exists)
> >
> > We will say that x is an unfindable natural
> > number.
> >
> > It is interesting to note that WM agrees with
> > the usual results if you insert the term findable.
> >
> > E.g.
> >
> > There is no findable last element of the potentially
> > infinite set |N.
> >
> > There is no findable index to a line of L that
> > contains d.
> >
> > There is no ball with a findable index in the vase.
> >
> > etc.
> >
> > It does not really matter if nonfindable natural
> > numbers exist or not. They have no effect.
> >
> > I suggest we give WM a teddy bear marked unfindable.

>
> I suggest, William keeps abd comforts it until he can find the first
> line of L that is not capable of containing everthing that its
> predecessors contain.
>


As long as one has d, which DOES contain every line of L that is capable
of containing everthing that its predecessors contain, one does not need
such an L.

In standard math, d is just a sort of union of all L's and its existence
is required in such standard set theories as ZF and NBG.

Model: in ZF with the von Neumann naturals each of WM's "lines" is
merely modeled by the identity mapping on a nonempty natural, so every
line is a superset of any prior line. But in ZF, the union of any such
well-defined set of sets is itself a set so that union is d, the
identity function on N.

This works perfectly in ZF, so until WM can disprove ZF, he loses.
--