Scale (descriptive Set Theory)

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.

Formal definition

Given a pointset ''A'' contained in some product space
:$A\subseteq X=X_0\times X_1\times\ldots X_$
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
:$(\phi_n)_$
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
:$\forall n\forall x(\varphi_n(x)\leq\varphi_n(y) \iff S(n,x,y) \iff T(n,x,y))$
where ''S'' is in Γ and ''T'' is in the dual pointclass of Γ (that is, the complement of ''T'' is in Γ).Kechris and Moschovakis 2008:37, with harmless reworking 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

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

Notes

References

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{{DEFAULTSORT:Scale (Descriptive Set Theory)
Descriptive set theory