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.
Description
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 {0,1,2,...} of counting numbers can be followed up by adding a symbol ω, which is the smallest infinite ordinal. So the set {0,1,2,...,ω} counts "up to infinity". Some ordinals are successors of one that came just before, like the counting numbers {1,2,...} (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 (see Ordinal arithmetic), 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.
See also
External links
- Ordinal number @ Wikipedia.org
