# ordinal number

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

 $x=\{z\in S\mid z

(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 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.

Title ordinal number OrdinalNumber 2013-03-22 12:07:55 2013-03-22 12:07:55 yark (2760) yark (2760) 8 yark (2760) Definition msc 03E10 ordinal VonNeumannOrdinal