Order type
From Wikipedia, the free encyclopedia
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.
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.
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.
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}.
