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 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 whole numbers. 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. An ordinal number can only be used to describe the order type of a well ordered set and not the order type of a well ordered proper class. A well ordered set is a set is a set with a relation > such that \n* For any elements x and y, exactly one of these statements is true \n* x > y \n* y = x \n* y > x \n* For any elements x, y, z, if y > x and z>y, z > x \n* Every nonempty subset has a least element, that is, it has an element x such that no element that is < x is in the subset, where y < x is another way of saying x > y. Two well ordered sets have the same order type if and only if there's a bijection from one to the other that converts the relation in the first one to the relation in the second one. Ordinals are distinct from cardinal numbers, which are useful for saying how many objects are 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 describe the same cardinal (see Hilbert's grand hotel). Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although the addition and multiplication are not commutative. Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and to classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.

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