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.

Title	ordinal number
Canonical name	OrdinalNumber
Date of creation	2013-03-22 12:07:55
Last modified on	2013-03-22 12:07:55
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	8
Author	yark (2760)
Entry type	Definition
Classification	msc 03E10
Synonym	ordinal
Related topic	VonNeumannOrdinal