\begin{definition}[Definition:Inductive Class/General]
Let $A$ be a class.
Let $g: A \to A$ be a mapping on $A$.
Then $A$ is '''inductive under $g$''' {{iff}}:
{{begin-axiom}}
{{axiom | n = 1
        | lc= $A$ contains the empty set:
        | q = 
        | m = \quad \O \in A
}}
{{axiom | n = 2
        | lc= $A$ is closed under $g$:
        | q = \forall x
        | m = \paren {x \in A \implies \map g x \in A}
}}
{{end-axiom}}
\end{definition}