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.
