Principle of mathematical induction

The principle of mathematical induction says:

Let $P(n)$ be a statement for each $n\in\mathbb{N}$. If

1. (Base case) $P(0)$ is true, and
2. (Inductive step) for every $n\in\mathbb{N}$, $P(n)\Rightarrow P(n+1)$, then $P(n)$ is true for all $n\in\mathbb{N}$.

Induction is equivalent (over basic set theory) to the well-ordering principle for ℕ : every nonempty subset of $\mathbb{N}$ has a least element.