Mathematical Induction by Ear // Kulakla Matematiksel Tümevarım

               Forgive me, Turkish-speaking friends, if I have distorted the expression "a cânta după ureche" . Maybe, I don't know, it sounds like hell in English too.

             Everyone knows what Mathematical Induction is. Let $P(n)$ be a STATEMENT about the natural numbers. By STATEMENT we mean what in Logic is called PREDICATE. Predicates can be defined in first-order logic. I would express the axiom of induction like this:

$$P(0)\wedge \forall n(P(n)\rightarrow P(n+1))\Rightarrow \forall nP(n) \tag{I}$$

There are many debates on this topic. I understand that (I) is an axiom-schema. If we want to express (I) as a single axiom, that is,

$$\forall P(P(0)\wedge \forall n(P(n)\rightarrow P(n+1))\Rightarrow \forall nP(n))$$

then we go beyond first-order logic, because we apply the $\forall$ quantifier to a predicate.