Definition:Ordinal Sequence Defined by Transfinite Recursion

Definition

Let $\On$ denote the class of all ordinals.

Let $g$ be a mapping defined for all sets.

Let $c$ be a set.

Then there exists a unique $\On$-sequence $S_0, S_1, \dots, S_\alpha, \dots$ such that:

$(1)$	$:$		$\ds S_0 $	$\ds = $	$\ds c $
$(2)$	$:$	$\ds \forall \alpha \in \On:$	$\ds S_{\alpha + 1} $	$\ds = $	$\ds \map g {S_\alpha} $
$(3)$	$:$	$\ds \forall \lambda \in K_{II}:$	$\ds S_\lambda $	$\ds = $	$\ds \bigcup_{\alpha \mathop < \lambda} S_\alpha $

where $K_{II}$ denotes the class of all limit ordinals.

$\Box$

The $\On$-sequence $S_0, S_1, \dots, S_\alpha, \dots$ is referred to as:

the $\On$-sequence defined from $g$ and $c$ by transfinite recursion.

2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 5$ Transfinite recursion theorems: Theorem $5.7$: Note

\((1)\)	$:$		\(\ds S_0 \)	\(\ds = \)	\(\ds c \)
\((2)\)	$:$	\(\ds \forall \alpha \in \On:\)	\(\ds S_{\alpha + 1} \)	\(\ds = \)	\(\ds \map g {S_\alpha} \)
\((3)\)	$:$	\(\ds \forall \lambda \in K_{II}:\)	\(\ds S_\lambda \)	\(\ds = \)	\(\ds \bigcup_{\alpha \mathop < \lambda} S_\alpha \)