An ordinal number is a well ordered set $S$ such that, for every $x \in S$ , $$ x = \{z \in S \mid z < x\} $$ (where $<$ is the ordering relation on $S$ ).

It follows immediately from the definition that every ordinal is a transitive set. Also note that if $a,b\in S$ then we have $a<b$ if and only if $a\in b$ .

There is a theory of ordinal arithmetic which allows construction of various ordinals. For example, all the numbers $0$ , $1$ , $2$ , ... have natural interpretations as ordinals, as does the set of natural numbers (including $0$ ), which in this context is often denoted by $\omega$ , and is the first infinite ordinal.