Definition:Ordinal/Definition 3

Definition
An ordinal is a strictly well-ordered set $\struct {S, \prec}$ such that:


 * $\forall a \in S: S_a = a$

where $S_a$ is the initial segment of $S$ determined by $a$.

From the definition of an initial segment, and Ordering on Ordinal is Subset Relation, we have that:


 * $S_a = \set {x \in S: x \subsetneqq a}$

From Initial Segment of Ordinal is Ordinal we have that $S_a$ is itself an ordinal.

Also see

 * Equivalence of Definitions of Ordinal