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 ?