Order type

such that both f and its inverse are monotonic (preserving orders of elements).

The order type of a well-ordered set X is sometimes expressed as ord(X).

[1] The order type of the integers and rationals is usually denoted

The natural numbers have order type denoted by ω, as explained below.

Every well-ordered set is order-equivalent to exactly one ordinal number, by definition.

Order types thus often take the form of arithmetic expressions of ordinals.

Firstly, the order type of the set of natural numbers is ω.

For example, any countable such model has order type ω + (ω* + ω) ⋅ η. Secondly, consider the set V of even ordinals less than ω ⋅ 2 + 7: As this comprises two separate counting sequences followed by four elements at the end, the order type is With respect to their standard ordering as numbers, the set of rationals is not well-ordered.

Any countable totally ordered set can be mapped injectively into the rational numbers in an order-preserving way.

When the order is moreover dense and has no highest nor lowest element, there even exist a bijective such mapping.