Category:Properties of Class of All Ordinals

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This category contains pages concerning Properties of Class of All Ordinals:

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

$\On$ has the following properties:

Zero is an Ordinal

The natural number $0$ is an element of $\On$.

Successor of Ordinal is an Ordinal

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

Let $\alpha \in \On$ be an ordinal.

Then its successor set $\alpha^+ = \alpha \cup \set \alpha$ is also an ordinal.

Union of a Chain of Ordinals is an Ordinal

Let $C$ be a chain of elements of $\On$.

Then its union $\bigcup C$ is also an element of $\On$.

Superinduction Principle

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 $A$

That is, let $A$ be a superinductive class under the successor mapping.

Then $A$ contains all ordinals:

$\On \subseteq A$