On Thursday, February 27, 2014 2:02:54 PM UTC-5, John Gabriel wrote: > 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.
This was not a detailed proof -- just a very brief outline of what you would have to do. It would take thousands of lines of formal proof to do all this.