##### Wiktionary

**ordinal number**

n. 1 (context grammar English) A word that expresses the relative position of an item in an ordered sequence. 2 (context arithmetic English) A number used to denote position in a sequence. 3 (context mathematics English) A generalized kind of number to denote the length of a well-order on a set.

##### WordNet

##### Wikipedia

**Ordinal number (linguistics)**

Cardinal

one

two

three

four

1

2

3

4

Ordinal

first

second

third

fourth

1st

2nd

3rd

4th

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. (2d and 3d are not used in British English.) 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.

**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 the process of counting: labeling the objects with distinct whole numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order.

For finite collections of objects, the ordinal numbers are just the counting numbers. For infinite collections, there is more than one notion of "order", and the one that appropriately generalizes the "one-after-another" sense from finite sets is called well-ordering. To well order a set means to label the items in a one-after-another fashion, but also allowing some labels to follow infinite collections of objects. For example, the set } of counting numbers can be followed up by adding a symbol ω, which is the smallest infinite ordinal. So the set } counts "up to infinity". Some ordinals are successors of one that came just before, like the counting numbers } (called *successor ordinals*), but some ordinals, like the ordinal ω, are not successors of other ordinals, and these are called *limit ordinals*.

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). Like other kinds of numbers, ordinals can be added, multiplied, and exponentiated, although the addition and multiplication are not commutative.

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.