Also, should commutivity of the operations "*" and "+" be added as an
axiom?
|
| Instead, it is easy and sufficient to define addition
| and multiplication by specifying how they interact
| with the identity, with the successor operation, and
| with each other:
| (0 + a) = a for all a
| S(a + b) = (a + Sb) for all a,b
| (0 * a) = 0 for all a
| (a * Sb) = ((a * b) + b) for all a,b
| and that's all that need be said on the subject; the
| symmetric, associative, and distributive properties can be
| derived from there.
|