Ordinal number

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 labelling the objects with distinct whole numbers. Ordinal numbers are thus the "labels" needed to arrange infinite collections of objects in order. 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).

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.

Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated. The finite ordinals (and the finite cardinals) are the natural numbers: 0, 1, 2, …, since any two total orderings of a finite set are order isomorphic. The least infinite ordinal is ω, which is identified with the cardinal number . However, in the transfinite case, beyond ω, ordinals draw a finer distinction than cardinals on account of their order information. Whereas there is only one countably infinite cardinal, namely itself, there are uncountably many countably infinite ordinals, namely

Ordinal number (linguistics)

In linguistics, ordinal numbers are words representing position or rank in a sequential order. The order may be of size, importance, chronology, and so on. In English, they are adjectives such as third and tertiary.

They differ from cardinal numbers, which represent quantity.

Ordinal numbers may be written in English with numerals and letter suffixes: 1st, 2nd or 2d, 3rd or 3d, 4th, 11th, 21st, 101st, 477th, etc. In some countries, written dates omit the suffix, although it is nevertheless pronounced. For example: 5 November 1605 (pronounced "the fifth of November ... "); November 5, 1605, ("November Fifth ..."). When written out in full with "of", however, the suffix is retained: the 5th of November. In other languages, different ordinal indicators are used to write ordinal numbers.

In American Sign Language, the ordinal numbers first through ninth are formed with handshapes similar to those for the corresponding cardinal numbers with the addition of a small twist of the wrist.