ordinaliset

Ordinaliset is a term used in some branches of set theory and logic to denote a structured set of ordinals endowed with a chosen linear order. It is not a standard term with a single universally accepted definition, but in common use it refers to a pair (A, ≺) where A is a set of ordinals and ≺ is a total, well-founded order on A that is compatible with the natural order of ordinals in the sense that α ≺ β implies α < β for all α, β in A. In many treatments A is additionally required to be downward closed: if β ∈ A and α < β, then α ∈ A. Some authors also require A to be closed under certain ordinal operations, such as taking suprema of chains of length less than a fixed cardinal, or to be transitive as a set of ordinals.

Examples: Finite sets A = {0,1,...,n} with the usual order form simple ordinaliset: finite initial segments of ω. The set A = ω (the set of all finite ordinals) with the natural order is another example. More generally, any initial segment [0, γ) of an ordinal γ is a canonical ordinaliset under the normal order. The pair (A, ≺) where ≺ is the standard less-than relation restricted to A also gives a canonical ordinaliset.

Uses: Ordinaliset are convenient for talking about ordinal-indexed families (Xα)α∈A or for describing stage-by-stage constructions in transfinite recursion, where A records the available stages and ≺ orders them. They relate to the notion of transitive sets of ordinals and to the study of hierarchical structures in the cumulative hierarchy.

See also: ordinal, transitive set, well-order, cumulative hierarchy.