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, 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

A
⊆
X
=
X
0
×
X
1
×
…
X
m
−
1
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

(
ϕ
n
)
n
<
ω
with the following properties:

If the sequence

*x*_{i} is such that

*x*_{i} is an element of

*A* for each natural number

*i*, and

*x*_{i} converges to an element

*x*in the product space

*X*, and
for each natural number

*n* there is an ordinal λ

_{n} such that φ

_{n}(

*x*_{i})=λ

_{n} for all sufficiently large

*i*, then

*x* is an element of

*A*, and
for each

*n*, φ

_{n}(x)≤λ

_{n}.

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

∀
n
∀
x
(
φ
n
(
x
)
≤
φ
n
(
y
)
⟺
S
(
n
,
x
,
y
)
⟺
T
(
n
,
x
,
y
)
)
where *S* is in Γ and *T* is in the dual pointclass of Γ (that is, the complement of *T* is in Γ). 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*.

Note also that 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*)=∞).

*This section is yet to be written*
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.

*This section is yet to be written*