Principle of Induction for the Natural Numbers

Let $\mathbb{N}$ be the set of natural numbers, with successor map $S$ as in that definition. We take as an axiom the following principle of induction.

If $A\subseteq\mathbb{N}$ satisfies

1. $1\in A$, and
2. $S(n)\in A$ for every $n\in A$,

then $A=\mathbb{N}$.