Ordinal (von Neumann)

A set $\alpha$ is an ordinal when both of the following hold.

1. $\alpha$ is a transitive set: every element of $\alpha$ is also a subset of $\alpha$, that is $x \in \alpha \Rightarrow x \subseteq \alpha$.
2. The membership relation restricted to $\alpha$, namely $\{(x, y) \in \alpha \times \alpha : x \in y\}$, is a strict well-order of $\alpha$ (Well-order and well-ordered set): it is irreflexive, transitive as a relation, trichotomous on $\alpha$, and every nonempty subset of $\alpha$ has an $\in$-least element.

Ordinals are written with lowercase Greek letters, and for ordinals we set

$\alpha < \beta :\iff \alpha \in \beta, \qquad \alpha \le \beta :\iff (\alpha \in \beta \text{ or } \alpha = \beta).$

Write $0 := \emptyset$, which is an ordinal because both clauses hold vacuously, and write $\alpha^{+} := \alpha \cup \{\alpha\}$ for the successor of $\alpha$.

• Two different transitivities. Clause 1 is about the set $\alpha$: it contains all members of its members. Clause 2 asks in part that the relation $\in$ be transitive on $\alpha$. If every element of $\alpha$ is itself a transitive set then the relation $\in$ is transitive on $\alpha$; the converse holds given clause 1, and fails without it. For $\alpha = \{\{\{\emptyset\}\}\}$ the relation $\{(x, y) \in \alpha \times \alpha : x \in y\}$ is empty, hence vacuously transitive, yet the sole element $\{\{\emptyset\}\}$ of $\alpha$ is not a transitive set, since $\{\emptyset\} \in \{\{\emptyset\}\}$ while $\emptyset \notin \{\{\emptyset\}\}$. Under clause 1 the two readings do coincide, because then every member of a member of $\alpha$ again lies in $\alpha$, where the relation has something to say. Neither clause implies the other, and both are needed.
• Every ordinal is literally the set of all smaller ordinals. By clause 1 and the notation above, $\alpha = \{\beta : \beta < \alpha\}$, so the ordinals carry their own order relation as membership. This is von Neumann's device, and it is what makes it unnecessary to define an order type as an equivalence class of well-orders: apart from the class of the empty well-order, which is the singleton $\{\varnothing\}$, such a class is a proper class rather than a set, whereas the representative chosen here is always a set.
• The natural numbers are ordinals, and so is $\omega$. Each natural number is a transitive set (Every natural number is a transitive set and is not a member of itself), and on $\mathbb{N}$ membership coincides with the usual strict order, so each natural number and $\mathbb{N} = \omega$ itself (The natural numbers $\mathbb{N}$ (von Neumann)) satisfy both clauses. Both facts are proved in $\omega$ is the least limit ordinal, where they are needed; they are recorded here because they are the intended picture.
• The Axiom of Foundation is not used here. Some treatments define an ordinal as a transitive set linearly ordered by $\in$, which is equivalent to the definition above only in the presence of Foundation (The Axiom of Foundation: $\forall S\,(\exists t\,(t \in S) \to \exists s\,(s \in S \wedge \neg\exists u\,(u \in s \wedge u \in S)))$), since Foundation is what supplies the least element. The least element property is written into the definition instead, so nothing on this page depends on Foundation even though the library does state it as an axiom of ZFC.
• The definition is absolute in a strong sense: whether a set is an ordinal depends only on its members and the membership relation among them, with no reference to any ambient construction. That is why ordinals can be used to index constructions in any model of ZF without further hypotheses.