Axiom of infinity: What is an inductive set?

Question

This Axiom states that there exists an inductive set. But, what is the definition of an inductive set?

Answer 1

An inductive set is exactly a set which satisfies the definition required in the explicit formulation of the axiom. So in some places it can be more economical to define the notion of inductive sets separately and then just say that "there exists an inductive set" instead.

Definition. We say that $A$ is an inductive set if $\varnothing\in A$, and whenever $x\in A$ then $x\cup\{x\}\in A$ as well.

For example $V_\omega$ is an inductive set, as well $\omega$ is the smallest inductive set (it is a subset of any inductive set).