Definition:Inductive Class/General

Definition

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} $

2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 4$ A double induction principle and its applications: Definition $4.1$

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