ordinal number

An ordinal number is a well ordered set $S$ such that, for every $x\in S$,

(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 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.