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 or division from your assumptions. You don't just get to use algebra as you did. It must follow from the successor function.