In mathematics, especially in set theory, ordinals may be used to label the elements of any given well-ordered set (the smallest element being labeled 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 which is not a label for an element of the set. This "length" is called the order type of the set. For other meanings of mathematics or math, see mathematics (disambiguation). ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ... In mathematics, a well-order (or well-ordering) on a set S is a total order on S with the property that every non-empty subset of S has a least element in this ordering. ...

Ordinals represent equivalence classes of well-ordered sets where the equivalence relation is order-isomorphism. Such an ordinal is the order type of any set in the equivalence class. In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x ∈ X | x ~ a } The notion of equivalence classes is useful for constructing sets out...

More formally, the order type of a well-ordered set is the unique ordinal for which there is an order-preserving bijection between the ordinal and the well-ordered set. In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ... A bijective function. ...

For example, consider the set of even ordinals less than ω·2+7, which is:

{0, 2, 4, 6, ...; ω, ω+2, ω+4, ...; ω·2, ω·2+2, ω·2+4, ω·2+6}.

Its order type is ω·2+4, that is:

{0, 1, 2, 3, ...; ω, ω+1, ω+2, ...; ω·2, ω·2+1, ω·2+2, ω·2+3}.

See also

• Ordinal number
• Well-order
• Bijection