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.