Ordinal number

Uncommon card · No. 87149

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...

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