Ordinal number

In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, nth, etc.) aimed to extend enumeration to infinite sets. Usually Greek letters are used for ordinal number variables to help distinguish them from natural number variables.

A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used. To extend this process to various infinite sets, ordinal numbers are defined more generally as a linearly ordered class of numbers that include the natural numbers and have the property that every non-empty collection (set or proper class) of ordinals has a least or "smallest" element (this is needed for giving a meaning to "the least unused element"). This more general definition allows us to define an ordinal number ${\displaystyle \omega }$ (omega) to be the least element that is greater than every natural number, along with ordinal numbers ⁠${\displaystyle \omega +1}$⁠, ⁠${\displaystyle \omega +2}$⁠, etc., which are even greater than ⁠${\displaystyle \omega }$⁠.

The Zermelo–Fraenkel set theory asserts that, for any set of ordinals, there exists another ordinal greater than all of them. The answer to the question "What if that set is the set of all ordinals?" (the Burali-Forti paradox) is that the collection of all ordinals is not a set, but a proper class.

A linear order such that every non-empty subset has a least element is called a well-order. The axiom of choice implies that every set can be well-ordered. Given two well-ordered sets, one is isomorphic to an initial segment of the other, and the isomorphism is unique. This allows a unique ordinal to be associated with each well-ordered set, known as its order type.

Ordinal numbers are distinct from cardinal numbers, which measure the size of sets. Although the distinction between ordinals and cardinals is not this apparent on finite sets (one can go from one to the other just by counting labels), they are very different in the infinite case, where different infinite ordinals can correspond to sets having the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although none of these operations are commutative.

Ordinals were introduced by Georg Cantor in 1883 to accommodate infinite sequences and classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.