On Saturday, 3 May 2014 20:55:10 UTC+2, Virgil wrote:
> > You have not shown an undefinable object. >
> But I have shown the necessity for them.
No. We have the fact that in set theory the set of definable numbers is uncountable by the fact that it cannot be listed and is countable by the fact that all finite expressions, expressing numbers, relations, functions, mathematicians, and nonsense, form a countable set.
You cannot avoid this contradiction. You believe that you could avoid it by undefinable numbers, but that is simply wrong. With the same right you could claim that pink unicorns would avoid the contradiction.