Scales were originally isolated as a concept in the theory of uniformization,[1] but have found wide applicability in descriptive set theory, with applications such as establishing bounds on the possible lengths of wellorderings of a given complexity, and showing (under certain assumptions) that there are largest countable sets of certain complexities.
Given a pointset A contained in some product space where each Xk is either the Baire space or a countably infinite discrete set, we say that a norm on A is a map from A into the ordinal numbers.
A scale on A is a countably infinite collection of norms with the following properties: By itself, at least granted the axiom of choice, the existence of a scale on a pointset is trivial, as A can be wellordered and each φn can simply enumerate A.
To make the concept useful, a definability criterion must be imposed on the norms (individually and together).
Here "definability" is understood in the usual sense of descriptive set theory; it need not be definability in an absolute sense, but rather indicates membership in some pointclass of sets of reals.