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# Def/Ordinal number

Definition of Ordinal number: Suppose that $S$ is a set. Define a binary relation $\leq$ on $S$ by: $$s_1 \leq s_2 \mbox{ means that } s_1 = s_2 \mbox{ or } s_1 \in s_2.$$

A set $S$ is called an ordinal (or ordinal number) if the following two properties are satisfied:

• The pair $(S, \leq)$ is a well-ordered set.
• Every element of $S$ is also a subset of $S$.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Binary relation, Def/Well ordered, Def/Successor, State/Axiom of regularity, Def/Union

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