Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction: How to prove one of them? 
Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction

How to prove one of them ?
On Proofwiki there is an article proving the equivalence of the statements listed above. However the proof of an individual statement depends on the proof of one of the others, so in order to prove them all, one must prove one of them ?
I know we can prove the Principle of induction by assuming there is a smallets $n \in \mathbb N$ such that $P(n)$ doesnt hold and then get a contradiction, since we have already proved $P(0) \land P(n-1) \Rightarrow P(n)$, so $P(n)$ is indeed true.
However in this proof we are actually relying on the well-ordering principle on $\mathbb N$ ? Otherwise how can one assume there is a smallets $n\in \mathbb N$ such that $P(n)$ is false ?
 A: These are principles, not theorems.  
If one principle holds, then they all must hold. If one principle fails to hold, they all fail to hold.  Hence, the principles are equivalent. (This is why we see the biconditional $\iff$ being used.)
Put another way, what we have is the following true assertion:

Well-ordering principle on $\mathbb N \iff$ Principle of induction $\iff$ Principle of strong induction.

This assertion claims nothing about the truth value of any one of the principles: it claims only that if one principle is true, they all must be true, and if one principle is false, they all must be false.  
And hence, we adopt the well-ordering principle on $\mathbb N$ as an axiom, or else derived from the principle of induction, which is itself basic (taken to be true).
A: Yes, the point of the equivalence is that if $\mathbb{N}$ has one of the three listed properties, it has them all.
It's interesting to think about what it would take to prove that one of these (say, well-ordering) does in fact hold for $\mathbb{N}$. Which axioms do you plan to use? You have the axioms of arithmetic, and the fact that $\mathbb{N}$ is totally ordered -- but these can't be enough, because the positive rational numbers $\mathbb{Q}^+$ also satisfy these axioms, and are not well-ordered. You need something else -- for instance, an axiom stating that there are no natural numbers between 0 and 1, and then also an axiom that prohibits elements "after" the elements $1, 1+1, 1+1+1, \ldots$, etc. Since the axioms of arithmetic are not enough to distinguish the natural numbers from other number systems like $\mathbb{Q}^+$, some extra axioms -- well-ordering, or others equivalent to it -- must be added.
A: These are very basic properties of the natural numbers, so any proof will depend on your definition of (the set of) natural numbers. What is it?
