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In the mathematical discipline of [[descriptive set theory]], a '''scale''' is a certain kind of object defined on a [[set (mathematics)|set]] of [[point (mathematics)|point]]s in some [[Polish space]] (for example, a scale might be defined on a set of [[real number]]s).  Scales were originally isolated as a concept in the theory of [[uniformization (descriptive set theory)|uniformization]],<ref>Kechris and Moschovakis 2008:28</ref> but have found wide applicability in descriptive set theory, with applications such as establishing bounds on the possible lengths of [[wellordering]]s of a given complexity, and showing (under certain assumptions) that there are largest [[countable set]]s of certain complexities.
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==Formal definition==
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Given a pointset ''A'' contained in some product space
:<math>A\subseteq X=X_0\times X_1\times\ldots X_{m-1}</math>
where each ''X<sub>k</sub>'' is either the [[Baire space (set theory)|Baire space]] or a countably infinite discrete set, we say that a ''norm'' on ''A'' is a map from ''A'' into the [[ordinal number]]s.  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
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:<math>(\phi_n)_{n<\omega}</math>
with the following properties:
: If the sequence ''x<sub>i</sub>'' is such that
:: ''x<sub>i</sub>'' is an element of ''A'' for each natural number ''i'', and
:: ''x<sub>i</sub>'' converges to an element ''x''in the product space ''X'', and
:: for each natural number ''n'' there is an ordinal &lambda;<sub>''n''</sub> such that &phi;<sub>n</sub>(''x<sub>i</sub>'')=&lambda;<sub>''n''</sub> for all sufficiently large ''i'', then
:''x'' is an element of ''A'', and
:for each ''n'', &phi;<sub>n</sub>(x)&le;&lambda;<sub>''n''</sub>.<ref>Kechris and Moschovakis 2008:37</ref>
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 &phi;<sub>''n''</sub> 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 &phi;<sub>''n''</sub> themselves are not sets of reals, but the corresponding [[prewellordering]]s are (at least in essence).


The idea is that, for a given pointclass &Gamma;, we want the prewellorderings below a given point in ''A'' to be uniformly represented both as a set in &Gamma; and as one in the dual pointclass of &Gamma;, relative to the "larger" point being an element of ''A''. Formally, we say that the &phi;<sub>''n''</sub> form a '''&Gamma;-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
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:<math>\forall n\forall x(\varphi_n(x)\leq\varphi_n(y) \iff S(n,x,y) \iff T(n,x,y))</math>
where ''S'' is in &Gamma; and ''T'' is in the dual pointclass of &Gamma; (that is, the complement of ''T'' is in &Gamma;).<ref>Kechris and Moschovakis 2008:37, with harmless reworking</ref>  Note here that we think of &phi;<sub>''n''</sub>(''x'') as being &infin; whenever ''x''&notin;''A''; thus the condition &phi;<sub>''n''</sub>(''x'')&le;&phi;<sub>''n''</sub>(''y''), for ''y''&isin;''A'', also implies ''x''&isin;''A''.


Note also that the definition does ''not'' imply that the collection of norms is in the intersection of &Gamma; with the dual pointclass of &Gamma;.  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 &phi;<sub>''n''</sub>(''x'')&le;&phi;<sub>''n''</sub>(''y'')=&infin;).
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==Applications==
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:''This section is yet to be written''


==Scale property==
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The scale property is a strengthening of the [[prewellordering property]]. For pointclasses of a certain form, it implies that [[relation (mathematics)|relations]] in the given pointclass have a [[uniformization (set theory)|uniformization]] that is also in the pointclass.


==Periodicity==
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:''This section is yet to be written''
 
==Notes==
<references/>
 
==References==
* {{Citation| author=Moschovakis, Yiannis N. | title=Descriptive Set Theory | publisher=North Holland | year=1980 |isbn=0-444-70199-0}}
* {{Citation|last1=Kechris|first1=Alexander S. |last2=Moschovakis | first2=Yiannis N. |editor1-first=Alexander S.| editor1-last=Kechris|editor2-first=|editor2-last=[[Benedikt Löwe]] |editor3-first=John R.|editor3-last=Steel|title=Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I |publisher=Cambridge University Press |year=2008 |pages=28–74 |chapter=Notes on the theory of scales |isbn=978-0-521-89951-2}}
 
{{DEFAULTSORT:Scale (Descriptive Set Theory)}}
[[Category:Descriptive set theory]]

Latest revision as of 16:15, 2 November 2014

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