## Ordinal number |

In **ordinal number**, or **ordinal**, is one generalization of the concept of a

An ordinal number is used to describe the

- (
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

Whereas ordinals are useful for *ordering* the objects in a collection, they are distinct from

Ordinals were introduced by ^{[1]} to accommodate infinite sequences and to classify ^{[2]}

- ordinals extend the natural numbers
- definitions
- transfinite sequence
- transfinite induction
- arithmetic of ordinals
- ordinals and cardinals
- some "large" countable ordinals
- topology and ordinals
- downward closed sets of ordinals
- history
- see also
- notes
- references
- external links

A *size* of a *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,

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 *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).

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 ω^{2}. Further on, there will be ω^{3}, then ω^{4}, and so on, and ω^{ω}, then ω^{ωω}, then later ω^{ωωω}, and even later ε_{0} (_{1}