O.K.

Here is the best example of the diagonal process that I know.

This is in

__The Mathematical Experience __ by Davis & Hersh, p237.

Think of the set of all functions that map positive integers to positive integers.

Now think that we can list that set:

.

Then define

. We can show that

but it cannot be in the list.

If we admit the

*rule of the excluded middle* , then it is proven that

cannot be listed.

If you do not admit the

*rule of the excluded middle* then all of that is pointless.

We are speaking different languages. Our languages actually determine the realities in which we live (see L

Wittgenstein.