Strong Principle of Induction

Question

What is the proof of the Strong Principle of Induction? Why can we assume that a principle holds up to "n," least of all "n+1"? When can we make this assumption and when can we not?

Answer 1

Taking the domain of naturals as implicit, Strong induction is the principle that:$$P(0)~,~\forall n\Big(\big(\forall m\leq n: P(m)\big)\to P(n+1)\Big) \vdash \forall n\in\Bbb N:P(n)$$

If the predicate holds in the base case, and it is true that the predicate will hold for any natural is a logical consequence of it holding for all prior numbers , then the predicate shall hold for any natural number.

Why can we assume that a principle holds up to "n," least of all "n+1"?

The assumption you make is that $P(m)$ holds for all $m$ up to an arbitrary $n$.   You then seek to demonstrate, through some valid argument, that $P(n+1)$ will be a logical consequence of that assumption.   This forms a conditional proof that the implication is true for all $n$.