Definition:Ordinal/Definition 3

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


 * $\forall \beta \in \alpha: \alpha_\beta = \beta$

where $\alpha_\beta$ is the initial segment of $\alpha$ determined by $\beta$:


 * $\alpha_\beta = \set {x \in \alpha: x \subsetneqq \beta}$

From Initial Segment of Ordinal is Ordinal we have that $\alpha_\beta$ is itself an ordinal.

Also see

 * Equivalence of Definitions of Ordinal