Definition:Limit Ordinal/Definition 1

Definition

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.

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.

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers