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
Scales arose from the question of finding a definable uniformization for a relation of a given complexity. That is, given a relation R, and supposing that for every x there is some y such that xRy, we would like an actual definable function f such that f(x) picks out a particular value y for which xRy.
If a relation — say, between points in the Baire space (which for purposes of descriptive set theory is more or less equivalent to the real numbers) — is "sufficiently definable", then it will have a so-called Suslin representation, a representation in terms of trees. A Suslin representation for a relation R in turn allows giving a definable uniformization for R (with the tree as a parameter to the definition); given x, it suffices to follow the leftmost branch of the tree of attempts to find a y such that xRy.[citation needed]
Scales are closely related to Suslin representations. In fact, if a subset of the Baire space has a κ-scale (that is, a scale all of whose norms take values less than κ see the formal definition below), then it also has a κ-Suslin representation (that is, it can be represented by the infinite branches through a tree on κ×ω). Conversely, if a set has a κ-Suslin representation, then it is κω-Suslin.[2]
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
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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