ordinal number OrdinalNumber 2002-01-05 01:37:15 2009-05-07 18:05:31 Definition \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} 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 $0$, $1$, $2$, \ldots 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.