Properties of Class of All Ordinals/Superinduction Principle

Theorem

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

Let $A$ be a class which satisfies the following $3$ conditions:

$(1)$	$:$	$A$ contains the zero ordinal $0$:		$\ds 0 \in A $
$(2)$	$:$	$A$ is closed under successor mapping:	$\ds \forall \alpha:$	$\ds \paren {\alpha \in A \implies \alpha^+ \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 $A$

Then $A$ contains all ordinals:

$\On \subseteq A$

We note that the zero ordinal, denoted $0$, is identified as the empty set:

$0 \:= \O$

Hence by definition $A$ is indeed a superinductive class under the successor mapping.

By the definition of ordinal:

$\alpha$ is an ordinal if and only if $\alpha$ is an element of every superinductive class.

The result follows.

$\blacksquare$

\((1)\)	$:$	$A$ contains the zero ordinal $0$:		\(\ds 0 \in A \)
\((2)\)	$:$	$A$ is closed under successor mapping:	\(\ds \forall \alpha:\)	\(\ds \paren {\alpha \in A \implies \alpha^+ \in A} \)
\((3)\)	$:$	$A$ is closed under chain unions:	\(\ds \forall C:\)	\(\ds \bigcup C \in A \)	where $C$ is a chain of elements of $A$