Sequential discreteness and clopen-I-Boolean classes

1987 ◽  
Vol 52 (1) ◽  
pp. 232-242
Author(s):  
Randall Dougherty
Keyword(s):  
Final Section ◽  
Baire Space ◽  
Boolean Operations ◽  
Index Set ◽  
Borel Sets ◽  
Borel Hierarchy ◽  
Wadge Reducibility ◽  
Analytic Sets ◽  
Converse Results

Kantorovich and Livenson [6] initiated the study of infinitary Boolean operations applied to the subsets of the Baire space and related spaces. It turns out that a number of interesting collections of subsets of the Baire space, such as the collection of Borel sets of a given type (e.g. the Fσ sets) or the collection of analytic sets, can be expressed as the range of an ω-ary Boolean operation applied to all possible ω-sequences of clopen sets. (Such collections are called clopen-ω-Boolean.) More recently, the ranges of I-ary Boolean operations for uncountable I have been considered; specific questions include whether the collection of Borel sets, or the collection of sets at finite levels in the Borel hierarchy, is clopen-I-Boolean.The main purpose of this paper is to give a characterization of those collections of subsets of the Baire space (or similar spaces) that are clopen-I-Boolean for some I. The Baire space version can be stated as follows: a collection of subsets of the Baire space is clopen-I-Boolean for some I iff it is nonempty and closed downward and σ-directed upward under Wadge reducibility, and in this case we may take I = ω2. The basic method of proof is to use discrete subsets of spaces of the form K2 to put a number of smaller clopen-I-Boolean classes together to form a large one. The final section of the paper gives converse results indicating that, at least in some cases, ω2 cannot be replaced by a smaller index set.

Boolean operations, Borel sets, and Hausdorff's question

1996 ◽  
Vol 61 (4) ◽  
pp. 1287-1304
Author(s):  
Abhijit Dasgupta
Keyword(s):  
Polish Space ◽  
Descriptive Set Theory ◽  
Boolean Operation ◽  
Boolean Operations ◽  
Borel Sets ◽  
Systematic Fashion ◽  
Borel Hierarchy ◽  
The Difference ◽  
Projective Hierarchy ◽  
Descriptive Set

The study of infinitary Boolean operations was undertaken by the early researchers of descriptive set theory soon after Suslin's discovery of the important operation. The first attempt to lay down their theory in a systematic fashion was the work of Kantorovich and Livenson [5], where they call these the analytical operations. Earlier, Hausdorff had introduced the δs operations — essentially same as the monotoneω-ary Boolean operations, and Kolmogorov, independently of Hausdorff, had discovered the same objects, which were used in his study of the R operator.The ω-ary Boolean operations turned out to be closely related to most of the classical hierarchies over a fixed Polish space X, including, e. g., the Borel hierarchy (), the difference hierarchies of Hausdorff (Dη()), the C-hierarchy (Cξ) of Selivanovski, and the projective hierarchy (): for each of these hierarchies, every level can be expressed as the range of an ω-ary Boolean operation applied to all possible sequences of open subsets of X. In the terminology of Dougherty [3], every level is “open-ω-Boolean” (if and are collections of subsets of X and I is any set, is said to be -I-Boolean if there exists an I-ary Boolean operation Φ such that = Φ, i. e. is the range of Φ restricted to all possible I-sequences of sets from ). If in addition, the space X has a basis consisting of clopen sets, then the levels of the above hierarchies are also “clopen-ω-Boolean.”

WADGE HIERARCHY OF DIFFERENCES OF CO-ANALYTIC SETS

2016 ◽  
Vol 81 (1) ◽  
pp. 201-215 ◽  
Author(s):  
KEVIN FOURNIER
Keyword(s):  
Baire Space ◽  
Borel Sets ◽  
Wadge Hierarchy ◽  
Analytic Sets ◽  
Image Position

AbstractWe begin the fine analysis of nonBorel pointclasses. Working in ZFC + DET$\left( {_1^1 } \right)$, we describe the Wadge hierarchy of the class of increasing differences of co-analytic subsets of the Baire space by extending results obtained by Louveau ([5]) for the Borel sets.

λ-scales, κ-Souslin sets and a new definition of analytic sets

1976 ◽  
Vol 41 (2) ◽  
pp. 373-378
Author(s):  
Douglas R. Busch
Keyword(s):  
Borel Sets ◽  
Analytic Sets ◽  
Definition Of

SummaryThe notions of sets of reals being κ-Souslin (κ a cardinal) and admitting a λ-scale (λ an ordinal) are due respectively to D. A. Martin and Y.N. Moschovakis. A set is ω-Souslin if and only if it is Σ11 (analytic). We show that a set is ω-Souslin if and only if it admits an (ω + l)-scale. Jointly with Martin and Solovay we show that if κ is uncountable and has cofinality ω, then being κ-Souslin is equivalent to admitting a κ-scale. Our results together with those of Kechris give a new simultaneous characterization of Σ11 and Δ11 (Borel) sets (a set is Σ11 if it admits an (ω + 1)-scale and Δ11 if it admits an ω-scale) and determine completely the relation between the κ-Souslin sets and the sets admitting λ-scales.

scholarly journals Hilary Putnam on the philosophy of logic and mathematics

2018 ◽  
Vol 33 (2) ◽  
pp. 183
Author(s):  
José Miguel Sagüillo Fernández-Vega
Keyword(s):  
Final Section ◽  
Mathematical Practice ◽  
Logical Truth ◽  
Hilary Putnam ◽  
Philosophy Of Logic ◽  
Logical Validity ◽  
And Mathematics

I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic conception of the working mathematician.

scholarly journals O ENSINO E A PESQUISA SOBRE POLÍTICA EXTERNA NO CAMPO DAS RELAÇÕES INTERNACIONAIS DO BRASIL

2012 ◽  
Vol 1 (2) ◽  
pp. 95-128 ◽  
Author(s):  
Carlos Aurélio Pimenta de Faria
Keyword(s):  
Social Sciences ◽  
Foreign Policy ◽  
International Relations ◽  
Final Section ◽  
Graduate Programs ◽  
Fourth Section ◽  
The Third ◽  
The Status ◽  
Theses And Dissertations

The purpose of this article is to analyze teaching and research on foreign policy in Brazil in the last two decades. The first section discusses how the main narratives about the evolution of International Relations in Brazil, considered as an area of knowledge, depict the place that has been designed, in the same area, to the study of foreign policy. The second section is devoted to an assessment of the status of foreign policy in IR teaching in the country, both at undergraduate and scricto sensu graduate programs. There is also a mapping and characterization of theses and dissertations which had foreign policy as object. The third section assesses the space given to studies on foreign policy in three academic forums nationwide, namely: the meetings of ABRI (Brazilian Association of International Relations), the ABCP (Brazilian Association of Political Science) and ANPOCS (National Association of Graduate Programs and Research in Social Sciences). In the fourth section there is a mapping and characterization of the published articles on foreign policy between 1990 and 2010, in the following IR Brazilian journals: Cena Internacional, Contexto Internacional, Política Externa and Revista Brasileira de Política Internacional. At last, the fifth and final section seeks to assess briefly the importance that comparative studies have in the sub-area of foreign policy in the country. The final considerations make a general assessment of the empirical research presented in the previous sections.

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).

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.

Part 1 Constitutionalism and Islam: Conceptual Issues, 1.1 Constitutionalism in Islamic Countries: A Contemporary Perspective of Islamic Law

2012 ◽  
Author(s):  
Kamali Mohammad Hashim
Keyword(s):  
Constitutional Law ◽  
Final Section ◽  
Islamic Law ◽  
General Characterization ◽  
The Third ◽  
Islamic Countries ◽  
Contemporary Perspective ◽  
Islam And Secularism ◽  
The Relationship

This chapter begins with a brief characterization of Islamic constitutional law and its underdeveloped status as compared with other branches of Islamic law. It then highlights salient differences between the Islamic and Western approaches to constitutional law and briefly discusses Islam and secularism. The next section provides a general characterization of the Islamic system of rule under four sub-headings. The first of these defines government in Islam as a trust (amānah); the second describes it as a limited and thus non-totalitarian government; the third addresses the Islamic system of rule as a qualified democracy; and the last characterizes it as a civilian not a theocratic system of government. The final section summarizes the main results of the preceding analysis and offers some tentative conclusions on the relationship between Islamic government and democratic constitutionalism.

scholarly journals On Almost Continuous Mappings and Baire Spaces

1978 ◽  
Vol 21 (2) ◽  
pp. 183-186 ◽  
Author(s):  
Shwu-Yeng T. Lin ◽  
You-Feng Lin
Keyword(s):  
Topological Space ◽  
Dense Subset ◽  
Baire Space ◽  
Real Numbers ◽  
Range Space ◽  
Continuous Mappings ◽  
Almost Continuous

AbstractIt is proved, in particular, that a topological space X is a Baire space if and only if every real valued function f: X →R is almost continuous on a dense subset of X. In fact, in the above characterization of a Baire space, the range space R of real numbers may be generalized to any second countable, Hausdorfï space that contains infinitely many points.

Index sets and degrees of unsolvability

1982 ◽  
Vol 47 (2) ◽  
pp. 241-248 ◽  
Author(s):  
Michael Stob
Keyword(s):  
Density Theorem ◽  
Index Set ◽  
Index Sets ◽  
Priority Method ◽  
Ingenious Method ◽  
Priority Argument ◽  
Degrees Of Unsolvability ◽  
Standard Techniques

The characterization of classes of r.e. sets by their index sets has proved valuable in producing new results about the r.e. sets and degrees. The classic example is Yates' proof [5, Theorem 7] of Sacks' density theorem for r.e. degrees using his classification of {e: We ≤TD) as Σ3(D) whenever D is r.e. Theorem 1 of this paper is a refinement of this index set theorem of Yates which has already proved to have interesting consequences about the r.e. degrees. This theorem was originally announced by Kallibekov [1, Theorem 1]. Kallibekov there proposed a new and ingenious method for doing priority arguments which has also since been used by Kinber [2]. Unfortunately his proof to this particular theorem contains an error. We have a totally different proof using standard techniques which is of independent interest.The proof to Theorem 1 is an infinite injury priority argument. In §1 therefore we give a short summary of the infinite injury priority method. We draw heavily on the exposition of Soare [4] where a complete description of the method is given along with many examples. In §2 we prove the main theorem and also give what we think are the most interesting corollaries to this theorem announced by Kallibekov. In §3 we prove a theorem about Σ3 sets of indices of r.e. sets. This theorem is a strengthening of a theorem of Kinber [2, Theorem 1] which was proved using a modification of Kallibekov's technique. As application, we use our theorem to show that an r.e. set A has supersets of every r.e. degree iff A is not simple.

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