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In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [1]
Ordinal indicator – Character(s) following an ordinal number (used when writing ordinal numbers, such as a super-script) Ordinal number – Generalization of "n-th" to infinite cases (the related, but more formal and abstract, usage in mathematics) Ordinal data, in statistics; Ordinal date – Date written as number of days since first day of ...
Ordinal addition on the natural numbers is the same as standard addition. The first transfinite ordinal is ω, the set of all natural numbers, followed by ω + 1, ω + 2, etc. The ordinal ω + ω is obtained by two copies
Transfinite numbers: Numbers that are greater than any natural number. Ordinal numbers: Finite and infinite numbers used to describe the order type of well-ordered sets. Cardinal numbers: Finite and infinite numbers used to describe the cardinalities of sets.
Every well-ordered set is order-equivalent to exactly one ordinal number, by definition. The ordinal numbers are taken to be the canonical representatives of their classes, and so the order type of a well-ordered set is usually identified with the corresponding ordinal. Order types thus often take the form of arithmetic expressions of ordinals.
aleph-nought, aleph-zero, or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal.The set of all finite ordinals, called or (where is the lowercase Greek letter omega), also has cardinality .
Since the epsilon numbers are an unbounded subclass of the ordinal numbers, they are enumerated using the ordinal numbers themselves. For any ordinal number , is the least epsilon number (fixed point of the exponential map) not already in the set {<}. It might appear that this is the non-constructive equivalent of the constructive definition ...
Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ω 1 {\displaystyle \omega _{1}} is often written as [ 0 , ω 1 ) {\displaystyle [0,\omega _{1})} , to emphasize that it is the space consisting of all ordinals smaller than ω 1 {\displaystyle \omega _{1}} .