Paradoxes of Set Theory - Basics - Ordinal Numbers

Ordinal Numbers

Besides the cardinality, which describes the size of a set, ordered sets also form a subject of set theory. The axiom of choice guarantees that every set can be well-ordered, which means that a total order can be imposed on its elements such that every nonempty subset has a first element with respect to that order. The order of a well-ordered set is described by an ordinal number. For instance, 3 is the ordinal number of the set {0, 1, 2} with the usual order 0 < 1 < 2; and ω is the ordinal number of the set of all natural numbers ordered the usual way. Neglecting the order, we are left with the cardinal number |N| = |ω| = .

Ordinal numbers can be defined with the same method used for cardinal numbers. Define two well-ordered sets to have the same order type by: there exists a bijection between the two sets respecting the order: smaller elements are mapped to smaller elements. Then an ordinal number is, by definition, a class consisting of all well-ordered sets of the same order type. To have the same order type is an equivalence relation on the class of well-ordered sets, and the ordinal numbers are the equivalence classes.

Two sets of the same order type have the same cardinality. The converse is not true in general for infinite sets: it is possible to impose different well-orderings on the set of natural numbers that give rise to different ordinal numbers.

There is a natural ordering on the ordinals, which is itself a well-ordering. Given any ordinal α, one can consider the set of all ordinals less than α. This set turns out to have ordinal number α. This observation is used for a different way of introducing the ordinals, in which an ordinal is equated with the set of all smaller ordinals. This form of ordinal number is thus a canonical representative of the earlier form of equivalence class.