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

ω

{\displaystyle \omega }

(omega) to be the least element that is greater than every natural number, along with ordinal numbers ⁠

ω

+

1

{\displaystyle \omega +1}

⁠, ⁠

ω

+

2

{\displaystyle \omega +2}

⁠, etc., which are even greater than ⁠

ω

{\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.