descriptive set theory

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Descriptive Set Theory

Author : Yiannis N. Moschovakis
ISBN : 9780821848135
Genre : Mathematics
File Size : 28. 79 MB
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Descriptive Set Theory is the study of sets in separable, complete metric spaces that can be defined (or constructed), and so can be expected to have special properties not enjoyed by arbitrary pointsets. This subject was started by the French analysts at the turn of the 20th century, most prominently Lebesgue, and, initially, was concerned primarily with establishing regularity properties of Borel and Lebesgue measurable functions, and analytic, coanalytic, and projective sets. Its rapid development came to a halt in the late 1930s, primarily because it bumped against problems which were independent of classical axiomatic set theory. The field became very active again in the 1960s, with the introduction of strong set-theoretic hypotheses and methods from logic (especially recursion theory), which revolutionized it. This monograph develops Descriptive Set Theory systematically, from its classical roots to the modern ``effective'' theory and the consequences of strong (especially determinacy) hypotheses. The book emphasizes the foundations of the subject, and it sets the stage for the dramatic results (established since the 1980s) relating large cardinals and determinacy or allowing applications of Descriptive Set Theory to classical mathematics. The book includes all the necessary background from (advanced) set theory, logic and recursion theory.

Classical Descriptive Set Theory

Author : Alexander Kechris
ISBN : 9781461241904
Genre : Mathematics
File Size : 32. 69 MB
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Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text presents a largely balanced approach to the subject, which combines many elements of the different traditions. It includes a wide variety of examples, more than 400 exercises, and applications, in order to illustrate the general concepts and results of the theory.

Invariant Descriptive Set Theory

Author : Su Gao
ISBN : 158488794X
Genre : Mathematics
File Size : 74. 98 MB
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Presents Results from a Very Active Area of Research Exploring an active area of mathematics that studies the complexity of equivalence relations and classification problems, Invariant Descriptive Set Theory presents an introduction to the basic concepts, methods, and results of this theory. It brings together techniques from various areas of mathematics, such as algebra, topology, and logic, which have diverse applications to other fields. After reviewing classical and effective descriptive set theory, the text studies Polish groups and their actions. It then covers Borel reducibility results on Borel, orbit, and general definable equivalence relations. The author also provides proofs for numerous fundamental results, such as the Glimm–Effros dichotomy, the Burgess trichotomy theorem, and the Hjorth turbulence theorem. The next part describes connections with the countable model theory of infinitary logic, along with Scott analysis and the isomorphism relation on natural classes of countable models, such as graphs, trees, and groups. The book concludes with applications to classification problems and many benchmark equivalence relations. By illustrating the relevance of invariant descriptive set theory to other fields of mathematics, this self-contained book encourages readers to further explore this very active area of research.

The Descriptive Set Theory Of Polish Group Actions

Author : Howard Becker
ISBN : 0521576059
Genre : Mathematics
File Size : 25. 45 MB
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Research monograph on set theory by two of the world's leading researchers.

Descriptive Set Theory And Dynamical Systems

Author : M. Foreman
ISBN : 0521786444
Genre : Mathematics
File Size : 78. 42 MB
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In recent years there has been a growing interest in the interactions between descriptive set theory and various aspects of the theory of dynamical systems, including ergodic theory and topological dynamics. This volume, first published in 2000, contains a collection of survey papers by leading researchers covering a wide variety of recent developments in these subjects and their interconnections. This book provides researchers and graduate students interested in either of these areas with a guide to work done in the other, as well as with an introduction to problems and research directions arising from their interconnections.

Descriptive Set Theory And The Structure Of Sets Of Uniqueness

Author : A. S. Kechris
ISBN : 0521358116
Genre : Mathematics
File Size : 73. 94 MB
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To make this work accessible to logicians as well as set theorists and analysts, classical and modern theory of sets of uniqueness are covered as well as the relevant parts of descriptive set theory.

Banach Spaces And Descriptive Set Theory Selected Topics

Author : Pandelis Dodos
ISBN : 9783642121524
Genre : Mathematics
File Size : 50. 9 MB
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This volume deals with problems in the structure theory of separable infinite-dimensional Banach spaces, with a central focus on universality problems. This topic goes back to the beginnings of the field and appears in Banach's classical monograph. The novelty of the approach lies in the fact that the answers to a number of basic questions are based on techniques from Descriptive Set Theory. Although the book is oriented on proofs of several structural theorems, in the main text readers will also find a detailed exposition of numerous “intermediate” results which are interesting in their own right and have proven to be useful in other areas of Functional Analysis. Moreover, several well-known results in the geometry of Banach spaces are presented from a modern perspective.

Recursive Aspects Of Descriptive Set Theory

Author : Richard Mansfield
ISBN : UOM:39015015614145
Genre : Mathematics
File Size : 49. 96 MB
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Descriptive Set Theory And Definable Forcing

Author : Jindřich Zapletal
ISBN : 9780821834503
Genre : Mathematics
File Size : 85. 45 MB
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The subject of the book is the relationship between definable forcing and descriptive set theory. The forcing serves as a tool for proving independence of inequalities between cardinal invariants of the continuum. The analysis of the forcing from the descriptive point of view makes it possible to prove absoluteness theorems of the type ``certain forcings are the provably best attempts to achieve consistency results of certain syntactical form'' and others. There are connections to such fields as pcf theory, effective descriptive set theory, determinacy and large cardinals, Borel equivalence relations, abstract analysis, and others.

Descriptive Set Theory And Forcing

Author : Arnold W. Miller
ISBN : 9781107168060
Genre : Mathematics
File Size : 24. 12 MB
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Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. In this volume, the fourth publication in the Lecture Notes in Logic series, Miller develops the necessary features of the theory of descriptive sets in order to present a new proof of Louveau's separation theorem for analytic sets. While some background in mathematical logic and set theory is assumed, the material is based on a graduate course given by the author at the University of Wisconsin, Madison, and is thus accessible to students and researchers alike in these areas, as well as in mathematical analysis.

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