Anyways, as for the proof, it is already in the axioms.
- There is a natural number 0.
- Every natural number a has a natural number successor, denoted by S(a).
-There is no natural number whose successor is 0.
-Distinct natural numbers have distinct successors: if a ‚ b, then S(a) ‚ S(b).
- If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.)