Ordinal number

Representation of the ordinal numbers up to ωω. Each turn of the spiral represents one power of ω

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 natural 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

  • (Trichotomy) For any elements x and y, exactly one of these statements is true
    • x > y
    • y = x
    • y > x
  • (Transitivity) For any elements x, y, z, if x > y and y > z, then x > z
  • (Well-foundedness) Every nonempty subset has a least element, that is, it has an element x such that there is no other element y in the subset where x > y

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[1] to accommodate infinite sequences and to classify derived sets, which he had previously introduced in 1872 while studying the uniqueness of trigonometric series.[2]

Ordinals extend the natural numbers

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.

Whereas the notion of cardinal number is associated with a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets that are called well-ordered (so intimately linked, in fact, that some mathematicians make no distinction between the two concepts). A well-ordered set is a totally ordered set (given any two elements one defines a smaller and a larger one in a coherent way) in which there is no infinite decreasing sequence (however, there may be infinite increasing sequences); equivalently, every non-empty subset of the set has a least element. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labelled 0, the one after that 1, the next one 2, "and so on") and to measure the "length" of the whole set by the least ordinal that is not a label for an element of the set. This "length" is called the order type of the set.

Any ordinal is defined by the set of ordinals that precede it: in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the order type of the ordinals less than it, i.e., the ordinals from 0 (the smallest of all ordinals) to 41 (the immediate predecessor of 42), and it is generally identified as the set {0,1,2,…,41}. Conversely, any set S of ordinals that is downward-closed — meaning that for any ordinal α in S and any ordinal β < α, β is also in S — is (or can be identified with) an ordinal.

There are infinite ordinals as well: the smallest infinite ordinal is ω, which is the order type of the natural numbers (finite ordinals) and that can even be identified with the set of natural numbers (indeed, the set of natural numbers is well-ordered—as is any set of ordinals—and since it is downward closed it can be identified with the ordinal associated with it, which is exactly how ω is defined).

A graphical "matchstick" representation of the ordinal ω². Each stick corresponds to an ordinal of the form ω·m+n where m and n are natural numbers.

Perhaps a clearer intuition of ordinals can be formed by examining a first few of them: as mentioned above, they start with the natural numbers, 0, 1, 2, 3, 4, 5, … After all natural numbers comes the first infinite ordinal, ω, and after that come ω+1, ω+2, ω+3, and so on. (Exactly what addition means will be defined later on: just consider them as names.) After all of these come ω·2 (which is ω+ω), ω·2+1, ω·2+2, and so on, then ω·3, and then later on ω·4. Now the set of ordinals formed in this way (the ω·m+n, where m and n are natural numbers) must itself have an ordinal associated with it: and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω, and even later ε0 (epsilon nought) (to give a few examples of relatively small—countable—ordinals). This can be continued indefinitely far ("indefinitely far" is exactly what ordinals are good at: every time one says "and so on" when enumerating ordinals, it defines a larger ordinal). The smallest uncountable ordinal is the set of all countable ordinals, expressed as ω1.