In the mathematical discipline of descriptive set theory, a **scale** is a certain kind of object defined on a set of points in some Polish space (for example, a scale might be defined on a set of real numbers). 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.

## Formal definition

Given a pointset *A* contained in some product space

where each *X _{k}* 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. Each norm has an associated prewellordering, where one element of

*A*precedes another element if the norm of the first is less than the norm of the second.

A *scale* on *A* is a countably infinite collection of norms

with the following properties:

- If the sequence
*x*is such that_{i}*x*is an element of_{i}*A*for each natural number*i*, and*x*converges to an element_{i}*x*in the product space*X*, and- for each natural number
*n*there is an ordinal λ_{n}such that φ_{n}(*x*)=λ_{i}_{n}for all sufficiently large*i*, then

*x*is an element of*A*, and- for each
*n*, φ_{n}(x)≤λ_{n}.^{[2]}

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 φ_{n} 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. The norms φ_{n} themselves are not sets of reals, but the corresponding prewellorderings are (at least in essence).

The idea is that, for a given pointclass Γ, we want the prewellorderings below a given point in *A* to be uniformly represented both as a set in Γ and as one in the dual pointclass of Γ, relative to the "larger" point being an element of *A*. Formally, we say that the φ_{n} form a **Γ-scale on A** if they form a scale on

*A*and there are ternary relations

*S*and

*T*such that, if

*y*is an element of

*A*, then

where *S* is in Γ and *T* is in the dual pointclass of Γ (that is, the complement of *T* is in Γ).^{[3]} Note here that we think of φ_{n}(*x*) as being ∞ whenever *x*∉*A*; thus the condition φ_{n}(*x*)≤φ_{n}(*y*), for *y*∈*A*, also implies *x*∈*A*.

The definition does *not* imply that the collection of norms is in the intersection of Γ with the dual pointclass of Γ. This is because the three-way equivalence is conditional on *y* being an element of *A*. For *y* not in *A*, it might be the case that one or both of *S(n,x,y)* or *T(n,x,y)* fail to hold, even if *x* is in *A* (and therefore automatically φ_{n}(*x*)≤φ_{n}(*y*)=∞).

## Applications

*This section is yet to be written*

## Scale property

The scale property is a strengthening of the prewellordering property. For pointclasses of a certain form, it implies that relations in the given pointclass have a uniformization that is also in the pointclass.

## Periodicity

*This section is yet to be written*

## Notes

## References

- Moschovakis, Yiannis N. (1980),
*Descriptive Set Theory*, North Holland, ISBN 0-444-70199-0 - Kechris, Alexander S.; Moschovakis, Yiannis N. (2008), "Notes on the theory of scales", in Kechris, Alexander S.; Benedikt Löwe; Steel, John R. (eds.),
*Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I*, Cambridge University Press, pp. 28–74, ISBN 978-0-521-89951-2