Equivalence of Definitions of Limit Ordinal

Theorem

The following definitions of the concept of Limit Ordinal are equivalent:

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.

Proof

Let $x \in \On$ be an element of the class of all ordinals.

From Categories of Elements under Well-Ordering, $x$ falls into one of the following $3$ categories:

$(1): \quad x$ is the smallest element of $\On$ with respect to the subset relation

$(2): \quad x$ is the immediate successor of another element $y \in \On$ with respect to the subset relation

$(3): \quad x$ is a limit element of $\On$ under the subset relation.

We have that Zero is Smallest Ordinal.

The result follows.

$\blacksquare$