On Wednesday, March 5, 2014 10:37:02 AM UTC-5, John Gabriel wrote: > Using my real axioms, one constructs the rational numbers from nothing and defines their properties and arithmetic. > > > > My axioms are based exactly on Euclid's Elements. > > > > Peano's "axioms" are a joke.
Such a joke that, these days, only internet math cranks consider Euclid as any kind of alternative to Peano.
But, if you want to lead some kind revival of ancient mathematics, you will have to formalize these "axioms" of yours and prove a critical number of theorems. Unfortunately for you, those theorems would have to include all of Peano's axioms. So, you would be no further ahead, but it would be an interesting exercise.
BTW, my DC Proof software could help you this endeavor. It allows you to formulate your own axioms, although most users would use those already built into the program. I myself have attempted to do such a formalization, but there are just too many gaps in your narrative. You would have to reasonably fill in those gaps based, if possible, on the ancient historical record, but it might be possible. Good luck.