Scale (descriptive set theory)
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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.
Motivation
- This section is yet to be written
Formal definition
- This section is yet to be written
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
- ^ Kechris and Moschovakis 2008:28
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.; 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.
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