§ Induction on natural numbers cannot be derived from other axioms

The idea is to consider a model of the naturals that obeys all axioms other than induction, and to then show how this model fails to be a model of induction. Thus, induction does not follow from the peano aximos minus the induction axiom. We build a model of naturals as MN{}M \equiv \mathbb N \cup \{ * \} where we define the successor on MM as succ(nN)=n+1succ(n \in \mathbb N) = n + 1 and succ()=succ(*) = *. Now let's try to prove P(m)succ(m)mP(m) \equiv succ(m) \neq m for all mMm \in M. P(0)P(0) holds as succ(0)=10succ(0) = 1 \neq 0. It is also true that if P(m)P(m), then P(m+1)P(m+1). However, it is NOT true that mM,P(m)\forall m \in M, P(m) since it does not hold for M* \in M. So we really do need induction as an axiom to rule out other things.