§ 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 $M \equiv \mathbb N \cup \{ * \}$ where we define the successor on $M$ as $succ(n \in \mathbb N) = n + 1$ and $succ(*) = *$. Now let's try to prove $P(m) \equiv succ(m) \neq m$ for all $m \in M$. $P(0)$ holds as $succ(0) = 1 \neq 0$. It is also true that if $P(m)$, then $P(m+1)$. However, it is NOT true that $\forall m \in M, P(m)$ since it does not hold for $* \in M$. So we really do need induction as an axiom to rule out other things.