Date: Feb 24, 2013 9:56 AM
Author: William Hughes
Subject: Re: Matheology § 222 Back to the roots
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.

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.