The Class Omega of Natural Numbers with Zero

In the setting of Class Theory NBG: the Axioms, Standing Conventions and Basic Notation:

¶ $\omega$ is the class

$$\omega=\{n:\forall y\,(y\text{ is inductive}\Rightarrow n\in y)\}$$

of all sets that are elements of every inductive set, formed by class abstraction; here $y$ is a set variable and "$y$ is inductive" is the defined predicate of Inductive Classes §inductive, so the formula is predicative as Class Theory NBG: the Axioms, Standing Conventions and Basic Notation §comprehension requires. We also write $\mathbb{N}_{0}$ for $\omega$.

¶ $0$ denotes the empty set: $0=\emptyset$.