sets of reals

1997 ◽  
Vol 62 (4) ◽  
pp. 1379-1428 ◽  
Author(s):  
Joan Bagaria ◽  
W. Hugh Woodin
Keyword(s):  
Set Theory ◽  
Real Numbers ◽  
Borel Sets ◽  
Definable Sets ◽  
Analytic Sets ◽  
Basic Facts ◽  
Continuous Images

Some of the most striking results in modern set theory have emerged from the study of simply-definable sets of real numbers. Indeed, simple questions like: what are the posible cardinalities?, are they measurable?, do they have the property of Baire?, etc., cannot be answered in ZFC.When one restricts the attention to the analytic sets, i.e., the continuous images of Borel sets, then ZFC does provide an answer to these questions. But this is no longer true for the projective sets, i.e., all the sets of reals that can be obtained from the Borel sets by taking continuous images and complements. In this paper we shall concentrate on particular projective classes, the , and using forcing constructions we will produce models of ZFC where, for some n, all , sets have some specified property. For the definition and basic facts about the projective classes , and , as well as the Kleene (or lightface) classes , and , we refer the reader to Moschovakis [19].The first part of the paper is about measure and category. Early in this century, Luzin [16] and Luzin-Sierpiński [17] showed that all analytic (i.e., ) sets of reals are Lebesgue measurable and have the property of Baire.

Analytic sets having incomparable kleene degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 860-868 ◽  
Author(s):  
Galen Weitkamp
Keyword(s):  
Set Theory ◽  
Characteristic Functions ◽  
Primary Concern ◽  
Descriptive Set Theory ◽  
Upper Semilattice ◽  
Definable Sets ◽  
Analytic Sets ◽  
Countably Infinite ◽  
Descriptive Set

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.

scholarly journals New Directions in Descriptive Set Theory

1999 ◽  
Vol 5 (2) ◽  
pp. 161-174 ◽  
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.

Two Applications of Inner Model Theory to the Study of Sets

1996 ◽  
Vol 2 (1) ◽  
pp. 94-107 ◽  
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.

scholarly journals A system of axiomatic set theory. Part III. Infinity and enumerability. Analysis

1942 ◽  
Vol 7 (2) ◽  
pp. 65-89 ◽  
Author(s):  
Paul Bernays
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Real Numbers ◽  
Full Generality ◽  
Axiomatic Set Theory

The foundation of analysis does not require the full generality of set theory but can be accomplished within a more restricted frame. Just as for number theory we need not introduce a set of all finite ordinals but only a class of all finite ordinals, all sets which occur being finite, so likewise for analysis we need not have a set of all real numbers but only a class of them, and the sets with which we have to deal are either finite or enumerable.We begin with the definitions of infinity and enumerability and with some consideration of these concepts on the basis of the axioms I—III, IV, V a, V b, which, as we shall see later, are sufficient for general set theory. Let us recall that the axioms I—III and V a suffice for establishing number theory, in particular for the iteration theorem, and for the theorems on finiteness.

Compositions of Set Operations

1970 ◽  
Vol 22 (2) ◽  
pp. 227-234
Author(s):  
D. W. Bressler ◽  
A. H. Cayford
Keyword(s):  
Topological Structure ◽  
Borel Sets ◽  
Set Operations ◽  
The Family ◽  
Analytic Sets

The set operations under consideration are Borel operations and Souslin's operation (). With respect to a given family of sets and in a setting free of any topological structure there are defined three Borel families (Definitions 3.1) and the family of Souslin sets (Definition 4.1). Conditions on an initial family are determined under which iteration of the Borel operations with Souslin's operation () on the initial family and the families successively produced results in a non-decreasing sequence of families of analytic sets (Theorem 5.2.1 and Definition 3.5). A classification of families of analytic sets with respect to an initial family of sets is indicated in a manner analogous to the familiar classification of Borel sets (Definition 5.3).

Fuzzy Logic in the Broad Sense

2017 ◽  
Author(s):  
Radim Bělohlávek ◽  
Joseph W. Dauben ◽  
George J. Klir
Keyword(s):  
Fuzzy Logic ◽  
Fuzzy Sets ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Human Reasoning ◽  
Human Beings ◽  
Real Numbers ◽  
New Directions ◽  
Gradual Development

The chapter begins by introducing the important and useful distinction between the research agendas of fuzzy logic in the narrow and the broad senses. The chapter deals with the latter agenda, whose ultimate goal is to employ intuitive fuzzy set theory for emulating commonsense human reasoning in natural language and other unique capabilities of human beings. Restricting to standard fuzzy sets, whose membership degrees are real numbers in the unit interval [0,1], the chapter describes how this broad agenda has become increasingly specific via the gradual development of standard fuzzy set theory and the associated fuzzy logic. An overview of currently recognized nonstandard fuzzy sets, which open various new directions in fuzzy logic, is presented in the last section of this chapter.

scholarly journals Linear metric spaces and analytic sets

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.

A minimal counterexample to universal baireness

1999 ◽  
Vol 64 (4) ◽  
pp. 1601-1627 ◽  
Author(s):  
Kai Hauser
Keyword(s):  
Fine Structure ◽  
Set Theory ◽  
Canonical Model ◽  
Core Model ◽  
Real Numbers ◽  
The Real ◽  
Minimal Counterexample ◽  
General Method ◽  
The Way

AbstractFor a canonical model of set theory whose projective theory of the real numbers is stable under set forcing extensions, a set of reals of minimal complexity is constructed which fails to be universally Baire. The construction uses a general method for generating non-universally Baire sets via the Levy collapse of a cardinal, as well as core model techniques. Along the way it is shown (extending previous results of Steel) how sufficiently iterable fine structure models recognize themselves as global core models.

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