Analytic sets having incomparable kleene degrees

One concern of descriptive set theory is the classification of definable sets of reals. Taken loosely ‘definable’ can mean anything from projective to formally describable in the language of Zermelo-Fraenkel set theory (ZF). Recursiveness, in the case of Kleene recursion, can be a particularly informative notion of definability. Sets of integers, for example, are categorized by their positions in the upper semilattice of Turing degrees, and the algorithms for computing their characteristic functions may be taken as their defining presentations. In turn it is interesting to position the common fauna of descriptive set theory in the upper semilattice of Kleene degrees. In so doing not only do we gain a perspective on the complexity of those sets common to the study of descriptive set theory but also a refinement of the theory of analytic sets of reals. The primary concern of this paper is to calculate the relative complexity of several notable coanalytic sets of reals and display (under suitable set theoretic hypothesis) several natural solutions to Post's problem for Kleene recursion.For sets of reals A and B one says A is Kleene recursive in B (written A ≤KB) iff there is a real y so that the characteristic function XA of A is recursive (in the sense of Kleene [1959]) in y, XB and the existential integer quantifier ∃; i.e. there is an integer e so that XA(x) ≃ {e}(x, y, XB, ≃). Intuitively, membership of a real x ϵ ωω in A can be decided from an oracle for x, y and B using a computing machine with a countably infinite memory and an ability to search and manipulate that memory in finite time. A set is Kleene semirecursive in B if it is the domain of an integer valued partial function recursive in y, XB and ≃ for some real y.

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Two Applications of Inner Model Theory to the Study of Sets

Bulletin of Symbolic Logic ◽

10.2307/421049 ◽

1996 ◽

Vol 2 (1) ◽

pp. 94-107 ◽

Cited By ~ 3

Author(s):

Greg Hjorth

Keyword(s):

Los Angeles ◽

Set Theory ◽

Model Theory ◽

Large Cardinals ◽

Descriptive Set Theory ◽

Inner Model Theory ◽

Regular Cardinals ◽

Inner Model ◽

Definable Sets ◽

Descriptive Set

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL(ℝ) are measurable. John Steel's analysis also settled a number of structural questions regarding HODL(ℝ), such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals.

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Linear metric spaces and analytic sets

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150000612x ◽

1994 ◽

Vol 37 (2) ◽

pp. 355-358

Author(s):

Robert Kaufman

Keyword(s):

Set Theory ◽

Convex Sets ◽

Metric Spaces ◽

Compact Convex ◽

Extreme Points ◽

Descriptive Set Theory ◽

Analytic Sets ◽

Descriptive Set

A problem in descriptive set theory, in which the objects of interest are compact convex sets in linear metric spaces, primarily those having extreme points.

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Descriptive set theory and harmonic analysis

Journal of Symbolic Logic ◽

10.2307/2275277 ◽

1992 ◽

Vol 57 (2) ◽

pp. 413-441 ◽

Cited By ~ 6

Author(s):

A. S. Kechris ◽

A. Louveau

Keyword(s):

Harmonic Analysis ◽

Set Theory ◽

Main Concern ◽

Descriptive Set Theory ◽

Subsequent Work ◽

Regularity Properties ◽

Short Course ◽

Precise Analysis ◽

Definable Sets ◽

Descriptive Set

During the 1989 European ASL Summer Meeting in Berlin, the authors gave a series of eight lectures (short course) on the topic of the title. This survey article consists basically of the lecture notes for that course distributed to the participants of that conference. We have purposely tried in this printed version to preserve the informal style of the original notes.Let us say first a few things aboui the content of these lectures. Our aim has been to present some recent work in descriptive set theory and its applications to an area of harmonic analysis. Typical uses of descriptive set theory in analysis are most often through regularity properties of definable sets, like measurability, the property of Baire, capacitability, etc., which are used to show that certain problems have solutions that behave nicely. In the theory we will present, definability itself, in fact the precise analysis of the “definable complexity” of certain sets, will be the main concern. It will be through such knowledge that we will be able to infer important structural properties of various objects which will then be used to solve analysis problems.The first lecture provides a short historical introduction to the subject of uniqueness for trigonometric series, which is the area of harmonic analysis whose problems are the origin of this work. As is well known, it was Cantor who proved the first major result in this subject in 1870, and it was his subsequent work here that led him to the creation of set theory.

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ω-powers and descriptive set theory

Journal of Symbolic Logic ◽

10.2178/jsl/1129642123 ◽

2005 ◽

Vol 70 (4) ◽

pp. 1210-1232 ◽

Cited By ~ 4

Author(s):

Dominique Lecomte

Keyword(s):

Set Theory ◽

Finite Alphabet ◽

Descriptive Set Theory ◽

Wadge Hierarchy ◽

Analytic Sets ◽

Descriptive Set

AbstractWe study the sets of the infinite sentences constructible with a dictionary over a finite alphabet, from the viewpoint of descriptive set theory. Among others, this gives some true co-analytic sets. The case where the dictionary is finite is studied and gives a natural example of a set at level ω of the Wadge hierarchy.

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New Directions in Descriptive Set Theory

Bulletin of Symbolic Logic ◽

10.2307/421088 ◽

1999 ◽

Vol 5 (2) ◽

pp. 161-174 ◽

Cited By ~ 20

Author(s):

Alexander S. Kechris

Keyword(s):

Hilbert Space ◽

Set Theory ◽

Descriptive Set Theory ◽

Borel Sets ◽

Polish Spaces ◽

Definable Sets ◽

Metrizable Spaces ◽

In The Beginning ◽

Projective Hierarchy ◽

Descriptive Set

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are ℝn, ℂn, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2ℕ, the Baire space ℕℕ, the infinite symmetric group S∞, the unitary group (of the Hilbert space), the group of measure preserving transformations of the unit interval, etc.In this theory sets are classified in hierarchies according to the complexity of their definitions and the structure of sets in each level of these hierarchies is systematically analyzed. In the beginning we have the Borel sets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the corresponding Borel hierarchy ( sets). After this come the projective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the corresponding projective hierarchy ( sets).There are also transfinite extensions of the projective hierarchy and even much more complex definable sets studied in descriptive set theory, but I will restrict myself here to Borel and projective sets, in fact just those at the first level of the projective hierarchy, i.e., the Borel (), analytic () and coanalytic () sets.

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