Definition:Superinductive Class

Definition

Let $A$ be a class.

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

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

That is:

$(1)$	$:$	$A$ contains the empty set:		$\ds \O \in A $
$(2)$	$:$	$A$ is closed under $g$:	$\ds \forall x:$	$\ds \paren {x \in A \implies \map g x \in A} $
$(3)$	$:$	$A$ is closed under chain unions:	$\ds \forall C:$	$\ds \bigcup C \in A $	where $C$ is a chain of elements of $A$

2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $4$: Superinduction, Well Ordering and Choice: Part $\text I$ -- Superinduction and Well Ordering: $\S 2$ Superinduction and double superinduction

\((1)\)	$:$	$A$ contains the empty set:		\(\ds \O \in A \)
\((2)\)	$:$	$A$ is closed under $g$:	\(\ds \forall x:\)	\(\ds \paren {x \in A \implies \map g x \in A} \)
\((3)\)	$:$	$A$ is closed under chain unions:	\(\ds \forall C:\)	\(\ds \bigcup C \in A \)	where $C$ is a chain of elements of $A$