# 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:

- $\On \subseteq A$

## Subcategories

This category has only the following subcategory.

### S

## Pages in category "Properties of Class of All Ordinals"

The following 8 pages are in this category, out of 8 total.