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.

Motivation

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Formal definition

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Applications

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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

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References

• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0.
• Kechris, Alexander S.; Moschovakis, Yiannis N. (1980). "Notes on the theory of scales". In Kechris, Alexander S.; Löwe, Benedikt; 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.