Definition:Inductive Class/General

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Let $A$ be a class.

Let $g: A \to A$ be a mapping on $A$.

Then $A$ is inductive under $g$ if and only if:

\((1)\)   $:$   $A$ contains the empty set:    \(\ds \quad \O \in A \)      
\((2)\)   $:$   $A$ is closed under $g$:      \(\ds \forall x:\) \(\ds \paren {x \in A \implies \map g x \in A} \)      

Also see

  • Results about inductive classes can be found here.