# Definition:Limit Ordinal

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## Definition

### Definition 1

An ordinal $\lambda$ is a **limit ordinal** if and only if it is a limit element in the well-ordering on the class of all ordinals $\On$ that is the subset relation.

### Definition 2

An ordinal $\lambda$ is a **limit ordinal** if and only if it is neither the zero ordinal nor a successor ordinal.

## Notation

The class of all **non-limit ordinals** can be denoted $K_I$, while the class of all **limit ordinals** can be denoted $K_{II}$.

## Also defined as

Some sources also consider the zero ordinal a **limit ordinal**.

It's a matter of taste.

## Also see

- Equivalence of Definitions of Limit Ordinal
- Class of All Ordinals is Well-Ordered by Subset Relation

- Results about
**limit ordinals**can be found**here**.