The induction axiom in the theory of Peano Arithmetic (PA) is actually an axiom scheme such that for every formula $\phi(x,\bar{y})$ with free variables $x,\bar{y}$ ($\bar{y}$ being a string of variables) we include axiom $$ \forall \bar{y} (\phi(0,\bar{y}) \wedge \forall x (\phi(x,\bar{y})\rightarrow \phi(s(x),\bar{y}))\rightarrow \forall x \phi(x,\bar{y})). $$
We often refer to $\mathbb{N}$ as the standard model of PA, in the sense that any model that is not isomorphic to it is considered nonstandard. The fact that $\mathbb{N}$ is a model of PA can be used to proof by means of the compactness theorem that PA is not $\omega$-categorical.
I understand $\mathbb{N}$ to be a structure in the language of PA such that all PA axioms except possibly induction hold and where every element is image of $0$ after applying the successor function a finite number of times.
How do we know that in $\mathbb{N}$ the induction axiom scheme holds?
A way of rephrasing the question could be how do we know that there exists a model of PA where every element is image of $0$ after taking successor finitely many times.