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.

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