On Friday, May 2, 2014 3:40:43 AM UTC-4, muec...@rz.fh-augsburg.de wrote:
> > No, he (Zermelo) has not. But his fear has come true. Set theory shows the uncountability of all countably many finite expressions. >
I really doubt that.
> > > > You might be able to construct an add function, but, from my experience, simple properties like associativity and commutativity seem to be impossible to establish. Addition, multiplication and exponentiation must be closed on the natural numbers. > > > > Of course. That is the problem pointing to neither finite nor actually infinite mathematics. Potential infinity appears to be the only resolution unless one is willing to consider MatheRealism. But that cannot be recommended as a general basis. >
After over a century of intensive scrutiny, no one has been able to demonstrate any inconsistency arising from that the Peano's Axioms for the infinite set of natural numbers. And no one, not even you WM, has been able to successfully develop number theory on any other formal basis.