§ 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
where we define the successor on as
and . Now let's try to prove for all .
holds as . It is also true that if , then . However,
it is NOT true that since it does not hold for . So we really do need
induction as an axiom to rule out other things.