What is missing from chx's description is the definition of the natural numbers (and the S (successor) function). The standard definition is: 0 is a natura

What is missing from chx's description is the definition of the natural numbers (and the S (successor) function). The standard definition is: 0 is a natural number S(n) is a natural number, if and only if n is a natural number x=y if and only if S(x)=S(y) for all x, S(x)!=0 These axioms are sufficient to show that the natural numbers form a line. At this point, we can use the definition of addition that chx provided: a+0=a a+S(b)=S(a+b) It is true that, without the restrictions on S, than defining a+b=0 would satisfy these. However, because of the restrictions on S from our definition of the natural numbers, it is impossible for S(a+b)=0.