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Author: Richard Mansfield
Publisher: Oxford University Press, USA
Published: 1985
Total Pages: 168
ISBN-13:
DOWNLOAD EBOOKExplores the nature of infinity with a view toward classifying and explaining its mathematical applications. It presents not only the basics of the classical theory, but also an introduction to the many important recent results and methods.
Author: Alexander Kechris
Publisher: Springer Science & Business Media
Published: 2012-12-06
Total Pages: 419
ISBN-13: 1461241901
DOWNLOAD EBOOKDescriptive 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.
Author: Howard Becker
Publisher: Cambridge University Press
Published: 1996-12-05
Total Pages: 152
ISBN-13: 0521576059
DOWNLOAD EBOOKIn this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has long been studied in many areas of mathematics. Non-locally compact Polish groups occur naturally as groups of symmetries in such areas as logic (especially model theory), ergodic theory, group representations, and operator algebras. Some of the topics covered here are: topological realizations of Borel measurable actions; universal actions; applications to invariant measures; actions of the infinite symmetric group in connection with model theory (logic actions); dichotomies for orbit spaces (including Silver, Glimm-Effros type dichotomies and the topological Vaught conjecture); descriptive complexity of orbit equivalence relations; definable cardinality of orbit spaces.
Author: Su Gao
Publisher: CRC Press
Published: 2008-09-03
Total Pages: 392
ISBN-13: 9781584887942
DOWNLOAD EBOOKPresents Results from a Very Active Area of ResearchExploring 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 mathem
Author: Yiannis N. Moschovakis
Publisher: American Mathematical Society
Published: 2025-01-31
Total Pages: 518
ISBN-13: 1470479877
DOWNLOAD EBOOKDescriptive 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.
Author: Arnold W. Miller
Publisher: Cambridge University Press
Published: 2017-05-18
Total Pages: 136
ISBN-13: 1316739317
DOWNLOAD EBOOKSince 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.
Author: Krzysztof Ciesielski
Publisher: Cambridge University Press
Published: 1997-08-28
Total Pages: 256
ISBN-13: 9780521594653
DOWNLOAD EBOOKPresents those methods of modern set theory most applicable to other areas of pure mathematics.
Author: Gerald E. Sacks
Publisher: Cambridge University Press
Published: 2017-03-02
Total Pages: 361
ISBN-13: 1107168430
DOWNLOAD EBOOKThis almost self-contained introduction to higher recursion theory is essential reading for all researchers in the field.
Author: Herbert B. Enderton
Publisher: Academic Press
Published: 1977-05-23
Total Pages: 294
ISBN-13: 0080570429
DOWNLOAD EBOOKThis is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning.
Author: S.M. Srivastava
Publisher: Springer
Published: 2013-12-01
Total Pages: 271
ISBN-13: 3642854737
DOWNLOAD EBOOKThe roots of Borel sets go back to the work of Baire [8]. He was trying to come to grips with the abstract notion of a function introduced by Dirich let and Riemann. According to them, a function was to be an arbitrary correspondence between objects without giving any method or procedure by which the correspondence could be established. Since all the specific functions that one studied were determined by simple analytic expressions, Baire delineated those functions that can be constructed starting from con tinuous functions and iterating the operation 0/ pointwise limit on a se quence 0/ functions. These functions are now known as Baire functions. Lebesgue [65] and Borel [19] continued this work. In [19], Borel sets were defined for the first time. In his paper, Lebesgue made a systematic study of Baire functions and introduced many tools and techniques that are used even today. Among other results, he showed that Borel functions coincide with Baire functions. The study of Borel sets got an impetus from an error in Lebesgue's paper, which was spotted by Souslin. Lebesgue was trying to prove the following: Suppose / : )R2 -- R is a Baire function such that for every x, the equation /(x,y) = 0 has a. unique solution. Then y as a function 0/ x defined by the above equation is Baire.
