On Thursday, 27 February 2014 20:50:09 UTC+2, Dan Christensen wrote: > On Thursday, February 27, 2014 1:09:00 PM UTC-5, John Gabriel wrote:
> 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."
Bollocks! You haven't defined subtraction and you haven't defined division nor multiplication from your assumptions.