Von Neumann definition of ordinals
| First few von Neumann ordinals | ||
|---|---|---|
| 0 | = Ø | |
| 1 | = { 0 } | = {Ø} |
| 2 | = { 0, 1 } | = { Ø, {Ø} } |
| 3 | = { 0, 1, 2 } | = { Ø, {Ø} , {Ø, {Ø}} } |
| 4 | = { 0, 1, 2, 3 } | = { Ø, {Ø}, {Ø, {Ø}}, {Ø, {Ø}, {Ø, {Ø}}} } |
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.
An ordinal number is used to describe the order type of a well ordered set (though this does not work for a well ordered proper class). A well ordered set is a set with a relation > such that
Two well ordered sets have the same order type if and only if there is a bijection from one set to the other that converts the relation in the first set to the relation in the second set.
Whereas ordinals are useful for ordering the objects in a collection, they 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. 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.
A natural number (which, in this context, includes the number 0) can be used for two purposes: to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets these two concepts coincide, there is only one way to put a finite set into a linear sequence, up to isomorphism. When dealing with infinite sets one has to distinguish between the notion of size, which leads to cardinal numbers, and the notion of position, which is generalized by the ordinal numbers described here. This is because while any set has only one size (its cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.
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