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
It is interesting to note that WM agrees with
the usual results if you insert the term findable.
There is no findable last element of the potentially
infinite set |N.
There is no findable index to a line of L that
There is no ball with a findable index in the vase.
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.