Successor Ordinal

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A successor ordinal, by definition, is an ordinal $\alpha$ that is equal to $\beta + 1$ for some ordinal $\beta$.

The successor function, denoted as $\beta+1$, $\operatorname{suc} \beta$, or $\beta ^+$ is defined on the ordinals as $\beta \cup \{ \beta \}$

Properties of the successor

There are no ordinals between $\beta$ and $\beta + 1$.

The union $\bigcup \beta$ can be thought of as an inverse successor function, because $\bigcup ( \beta + 1 ) = \beta$. All limit ordinals are equal to their union.