On Thursday, February 27, 2014 1:09:00 PM UTC-5, John Gabriel wrote: > > That's all very nice, but I prefer Peano's five, relatively succinct axioms as a starting point. You can probably derive some equivalent of all your "axioms" from them. And they have the pleasing property of describing the structure at least one subset of EVERY infinite set. > > > > Except Peano's axioms are a joke, and you've been had. Sorry, you CANNOT derive my sound axioms from that rot. Please do show me how. You are talking nonsense!
Briefly, you can construct the addition and multiplication functions from Peano's axioms such that:
x+0 = x x+S(y) = S(x+y)
x*0 = 0 x*S(y) = x*y + x
It's tricky working with recursively defined functions like this, but it can be done. From these definitions, you can derive the usual algebraic properties of addition and multiplication: associative, commutative, etc.
Then define subtraction and division as:
x-y = z <=> x = z + y x\y = z <=> x = z * y
Then you should be able to derive all of your "axioms of arithmetic."