In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly infinite) collection of objects in order, one after another.
Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. The basic idea of ordinal numbers is to generalize this process to possibly infinite collections and to provide a "label" for each step in the process. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order.
An ordinal number is used to describe the order type of a well-ordered set (though this does not work for a well-ordered proper class). A well-ordered set is a set with a relation < such that:
(Trichotomy) For any elements x and y, exactly one of these statements is true:
x < y
y < x
x = y
(Transitivity) For any elements x, y, z, if x < y and y < z, then x < z.
(Well-foundedness) Every nonempty subset has a least element, that is, it has an element x such that there is no other element y in the subset where y < x.Two well-ordered sets have the same order type, if and only if there is a bijection from one set to the other that converts the relation in the first set, to the relation in the second set.
Whereas ordinals are useful for ordering the objects in a collection, they are distinct from cardinal numbers, which are useful for quantifying the number of objects in a collection. Although the distinction between ordinals and cardinals is not always apparent in finite sets (one can go from one to the other just by counting labels), different infinite ordinals can correspond to the same cardinal. Moreover, there may be sets which cannot be well ordered, and their cardinal numbers do not correspond to ordinal numbers. (For example, the existence of such sets follows from Zermelo-Fraenkel set theory with the negation of the axiom of choice.) Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations is commutative.
Ordinals were introduced by Georg Cantor in 1883 in order to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872—while studying the uniqueness of trigonometric series.