It is axiomatized! It follows from the field axioms.
"Charlie-Boo" <email@example.com> wrote in message news://firstname.lastname@example.org... > Hello all, > > I am interested in axiomizing Number Theory. I'm not talking about > some bogus list of properties of addition and multiplication, but > rather a set of formal axioms and rules of inference that allows us (a > program) to derive theorems from Number Theory. > > It seems to be a natural branch of mathematics to axiomize, since it > figures so heavily in proofs concerning Logic (e.g. Godel's > Incompleteness Theorems) and Logic is so easily axiomized. I have > axiomized the Theory of Computation (a.k.a. Computability) and Program > Synthesis of Number Theoretic functions (prime listing or checking, > factoring, etc.) in http://www.mathpreprints.com/math/Preprint/CharlieVolkstorf/20021008.1/1 > and http://www.arxiv.org/html/cs.lo/0003071 . > > The first step is to gather together a few dozen of the very simplest > theorems from Number Theory. Then we look for primitives etc. > > The effort can be done here, or in private collaboration with me > (contact me at the email address below) and whoever else is > interested, with periodic progress reports posted here, culminating > with a published paper. > > Charlie Volkstorf > Cambridge, MA > axiomize at aol dot com