It is a familiar fact that the principle of mathematical induction and the principle of well-ordering are equivalent (in the set of natural numbers). However, the defining property of the natural numbers is the mathematical induction itself, i.e., the set of natural numbers is defined to be the smallest inductive set containing $1$ considering the usual axioms of set theory.
In what sense is the well-ordering principle equivalent to the mathematical induction? Or similarly, how can we define the natural numbers using the usual axioms of set theory minus the axiom of infinity (that there exist inductive sets) plus the principle of well-ordering?