Ordinal Number

An ordinal number is a mathematical object used to describe the position of an element within a well-ordered set, a set in which every non-empty subset has a smallest element, extending the everyday notion of first, second, third, and so on beyond any finite stopping point. The finite ordinals coincide with the ordinary natural numbers, but the first infinite ordinal, written as the lowercase Greek letter omega, comes immediately after every finite ordinal, and further ordinals such as omega plus one, omega times two, and omega raised to the power of omega continue the sequence indefinitely, each one representing a distinct and strictly larger way of well-ordering a set. Ordinal numbers were introduced by the German mathematician Georg Cantor in the 1880s as part of his development of set theory, growing directly out of his earlier work on infinite cardinal numbers, though ordinals and cardinals diverge once transfinite numbers are involved, since several different ordinals can correspond to sets of the very same cardinality. Ordinal numbers underlie the technique of transfinite induction, a method of proof that extends ordinary mathematical induction beyond the natural numbers, and they play a foundational role throughout modern set theory, including in the formal construction of the cumulative hierarchy of sets used in the standard Zermelo-Fraenkel axioms.