Set Theory

With an Introduction to Real Point Sets
Author: Abhijit Dasgupta
Publisher: Springer Science & Business Media
ISBN: 1461488540
Category: Mathematics
Page: 444
View: 3943

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What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. The core Cantor–Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin's problems on projective sets using determinacy of infinite games and large cardinals. Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study.

Introduction to the Theory of Sets


Author: Joseph Breuer
Publisher: Courier Corporation
ISBN: 0486154874
Category: Mathematics
Page: 128
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This undergraduate text develops its subject through observations of the physical world, covering finite sets, cardinal numbers, infinite cardinals, and ordinals. Includes exercises with answers. 1958 edition.

The Real Numbers

An Introduction to Set Theory and Analysis
Author: John Stillwell
Publisher: Springer Science & Business Media
ISBN: 331901577X
Category: Mathematics
Page: 244
View: 5675

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While most texts on real analysis are content to assume the real numbers, or to treat them only briefly, this text makes a serious study of the real number system and the issues it brings to light. Analysis needs the real numbers to model the line, and to support the concepts of continuity and measure. But these seemingly simple requirements lead to deep issues of set theory—uncountability, the axiom of choice, and large cardinals. In fact, virtually all the concepts of infinite set theory are needed for a proper understanding of the real numbers, and hence of analysis itself. By focusing on the set-theoretic aspects of analysis, this text makes the best of two worlds: it combines a down-to-earth introduction to set theory with an exposition of the essence of analysis—the study of infinite processes on the real numbers. It is intended for senior undergraduates, but it will also be attractive to graduate students and professional mathematicians who, until now, have been content to "assume" the real numbers. Its prerequisites are calculus and basic mathematics. Mathematical history is woven into the text, explaining how the concepts of real number and infinity developed to meet the needs of analysis from ancient times to the late twentieth century. This rich presentation of history, along with a background of proofs, examples, exercises, and explanatory remarks, will help motivate the reader. The material covered includes classic topics from both set theory and real analysis courses, such as countable and uncountable sets, countable ordinals, the continuum problem, the Cantor–Schröder–Bernstein theorem, continuous functions, uniform convergence, Zorn's lemma, Borel sets, Baire functions, Lebesgue measure, and Riemann integrable functions.

Introduction to Set Theory and Topology


Author: Kazimierz Kuratowski
Publisher: Elsevier
ISBN: 1483151638
Category: Mathematics
Page: 352
View: 6484

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Introduction to Set Theory and Topology describes the fundamental concepts of set theory and topology as well as its applicability to analysis, geometry, and other branches of mathematics, including algebra and probability theory. Concepts such as inverse limit, lattice, ideal, filter, commutative diagram, quotient-spaces, completely regular spaces, quasicomponents, and cartesian products of topological spaces are considered. This volume consists of 21 chapters organized into two sections and begins with an introduction to set theory, with emphasis on the propositional calculus and its application to propositions each having one of two logical values, 0 and 1. Operations on sets which are analogous to arithmetic operations are also discussed. The chapters that follow focus on the mapping concept, the power of a set, operations on cardinal numbers, order relations, and well ordering. The section on topology explores metric and topological spaces, continuous mappings, cartesian products, and other spaces such as spaces with a countable base, complete spaces, compact spaces, and connected spaces. The concept of dimension, simplexes and their properties, and cuttings of the plane are also analyzed. This book is intended for students and teachers of mathematics.

Set Theory and its Philosophy

A Critical Introduction
Author: Michael Potter
Publisher: Clarendon Press
ISBN: 0191556432
Category: Philosophy
Page: 360
View: 5250

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Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.

Classical Descriptive Set Theory


Author: Alexander Kechris
Publisher: Springer Science & Business Media
ISBN: 1461241901
Category: Mathematics
Page: 404
View: 5921

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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.

Elements of Set Theory


Author: Herbert B. Enderton
Publisher: Academic Press
ISBN: 0080570429
Category: Mathematics
Page: 279
View: 7171

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This 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.

Notes on Set Theory


Author: Yiannis N. Moschovakis
Publisher: Springer Science & Business Media
ISBN: 9783540941804
Category: Axiomatische Mengenlehre
Page: 272
View: 4512

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The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. It is also viewed as a foundation of mathematics so that "to make a notion precise" simply means "to define it in set theory." This book gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets, and also attempts to explain how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author has added solutions to the exercises, and rearranged and reworked the text to improve the presentation.

Problems and Theorems in Classical Set Theory


Author: Peter Komjath,Vilmos Totik
Publisher: Springer Science & Business Media
ISBN: 0387362193
Category: Mathematics
Page: 516
View: 7948

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This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution.

An Introduction to Measure Theory


Author: Terence Tao
Publisher: American Mathematical Soc.
ISBN: 0821869191
Category: Mathematics
Page: 206
View: 6715

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This is a graduate text introducing the fundamentals of measure theory and integration theory, which is the foundation of modern real analysis. The text focuses first on the concrete setting of Lebesgue measure and the Lebesgue integral (which in turn is motivated by the more classical concepts of Jordan measure and the Riemann integral), before moving on to abstract measure and integration theory, including the standard convergence theorems, Fubini's theorem, and the Caratheodory extension theorem. Classical differentiation theorems, such as the Lebesgue and Rademacher differentiation theorems, are also covered, as are connections with probability theory. The material is intended to cover a quarter or semester's worth of material for a first graduate course in real analysis. There is an emphasis in the text on tying together the abstract and the concrete sides of the subject, using the latter to illustrate and motivate the former. The central role of key principles (such as Littlewood's three principles) as providing guiding intuition to the subject is also emphasized. There are a large number of exercises throughout that develop key aspects of the theory, and are thus an integral component of the text. As a supplementary section, a discussion of general problem-solving strategies in analysis is also given. The last three sections discuss optional topics related to the main matter of the book.

Theory of Sets


Author: E. Kamke
Publisher: Courier Corporation
ISBN: 048645083X
Category: Mathematics
Page: 144
View: 706

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This introduction to the theory of sets employs the discoveries of Cantor, Russell, Weierstrass, Zermelo, Bernstein, Dedekind, and others. It analyzes concepts and principles, offering numerous examples. Topics include the rudiments of set theory, arbitrary sets and their cardinal numbers, ordered sets and their order types, and well-ordered sets and their ordinal numbers. 1950 edition.

From the Calculus to Set Theory, 1630-1910

An Introductory History
Author: I. Grattan-Guinness,H. J. M. Bos
Publisher: Princeton University Press
ISBN: 9780691070827
Category: Mathematics
Page: 306
View: 490

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From the Calculus to Set Theory traces the development of the calculus from the early seventeenth century through its expansion into mathematical analysis to the developments in set theory and the foundations of mathematics in the early twentieth century. It chronicles the work of mathematicians from Descartes and Newton to Russell and Hilbert and many, many others while emphasizing foundational questions and underlining the continuity of developments in higher mathematics. The other contributors to this volume are H. J. M. Bos, R. Bunn, J. W. Dauben, T. W. Hawkins, and K. Møller-Pedersen.

An Introduction to Algebraic Structures


Author: Joseph Landin
Publisher: Courier Corporation
ISBN: 0486150410
Category: Mathematics
Page: 272
View: 413

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This self-contained text covers sets and numbers, elements of set theory, real numbers, the theory of groups, group isomorphism and homomorphism, theory of rings, and polynomial rings. 1969 edition.

Introduction to Modern Mathematics


Author: Helena Rasiowa
Publisher: Elsevier
ISBN: 1483274721
Category: Mathematics
Page: 352
View: 4101

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Introduction to Modern Mathematics focuses on the operations, principles, and methodologies involved in modern mathematics. The monograph first tackles the algebra of sets, natural numbers, and functions. Discussions focus on groups of transformations, composition of functions, an axiomatic approach to natural numbers, intersection of sets, axioms of the algebra of sets, fields of sets, prepositional functions of one variable, and difference of sets. The text then takes a look at generalized unions and intersections of sets, Cartesian products of sets, and equivalence relations. The book ponders on powers of sets, ordered sets, and linearly ordered sets. Topics include isomorphism of linearly ordered sets, dense linear ordering, maximal and minimal elements, quasi-ordering relations, inequalities for cardinal numbers, sets of the power of the continuum, and Cantor's theorem. The manuscript then examines elementary concepts of abstract algebras, functional calculus and its applications in mathematical proofs, and propositional calculus and its applications in mathematical proofs. The publication is a valuable reference for mathematicians and researchers interested in modern mathematics.

The Theory of Sets of Points

Second Edition
Author: W. H. Young,Grace Chisholm Young
Publisher: American Mathematical Soc.
ISBN: 1470409623
Category:
Page: 326
View: 9957

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From the Preface to the First Edition (1906): ``There are no definitely accepted landmarks in the didactic treatment of Georg Cantor's magnificent theory, which is the subject of the present volume. A few of the most modern books on the Theory of Functions devote some pages to the establishment of certain results belonging to our subject, and required for the special purposes in hand ... But we may fairly claim that the present work is the first attempt at a systematic exposition of the subject as a whole.'' In this second edition, notes have been added by I. Grattan-Guinness drawn from extensive annotations in the author's own copy. A further appendix has been added.

Set Theory and the Continuum Hypothesis


Author: Paul J. Cohen,Martin Davis
Publisher: Courier Corporation
ISBN: 0486469212
Category: Mathematics
Page: 154
View: 8762

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This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

Introductory Real Analysis


Author: A. N. Kolmogorov,S. V. Fomin
Publisher: Courier Corporation
ISBN: 0486134741
Category: Mathematics
Page: 416
View: 8473

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Comprehensive, elementary introduction to real and functional analysis covers basic concepts and introductory principles in set theory, metric spaces, topological and linear spaces, linear functionals and linear operators, more. 1970 edition.

Spaces: An Introduction to Real Analysis


Author: Tom L. Lindstrøm
Publisher: American Mathematical Soc.
ISBN: 1470440628
Category: Functional analysis
Page: 369
View: 4566

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Spaces is a modern introduction to real analysis at the advanced undergraduate level. It is forward-looking in the sense that it first and foremost aims to provide students with the concepts and techniques they need in order to follow more advanced courses in mathematical analysis and neighboring fields. The only prerequisites are a solid understanding of calculus and linear algebra. Two introductory chapters will help students with the transition from computation-based calculus to theory-based analysis. The main topics covered are metric spaces, spaces of continuous functions, normed spaces, differentiation in normed spaces, measure and integration theory, and Fourier series. Although some of the topics are more advanced than what is usually found in books of this level, care is taken to present the material in a way that is suitable for the intended audience: concepts are carefully introduced and motivated, and proofs are presented in full detail. Applications to differential equations and Fourier analysis are used to illustrate the power of the theory, and exercises of all levels from routine to real challenges help students develop their skills and understanding. The text has been tested in classes at the University of Oslo over a number of years.

Introduction to Analysis


Author: Maxwell Rosenlicht
Publisher: Courier Corporation
ISBN: 0486134687
Category: Mathematics
Page: 272
View: 527

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Written for junior and senior undergraduates, this remarkably clear and accessible treatment covers set theory, the real number system, metric spaces, continuous functions, Riemann integration, multiple integrals, and more. 1968 edition.

An Introduction to Statistical Learning

with Applications in R
Author: Gareth James,Daniela Witten,Trevor Hastie,Robert Tibshirani
Publisher: Springer Science & Business Media
ISBN: 1461471389
Category: Mathematics
Page: 426
View: 9327

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An Introduction to Statistical Learning provides an accessible overview of the field of statistical learning, an essential toolset for making sense of the vast and complex data sets that have emerged in fields ranging from biology to finance to marketing to astrophysics in the past twenty years. This book presents some of the most important modeling and prediction techniques, along with relevant applications. Topics include linear regression, classification, resampling methods, shrinkage approaches, tree-based methods, support vector machines, clustering, and more. Color graphics and real-world examples are used to illustrate the methods presented. Since the goal of this textbook is to facilitate the use of these statistical learning techniques by practitioners in science, industry, and other fields, each chapter contains a tutorial on implementing the analyses and methods presented in R, an extremely popular open source statistical software platform. Two of the authors co-wrote The Elements of Statistical Learning (Hastie, Tibshirani and Friedman, 2nd edition 2009), a popular reference book for statistics and machine learning researchers. An Introduction to Statistical Learning covers many of the same topics, but at a level accessible to a much broader audience. This book is targeted at statisticians and non-statisticians alike who wish to use cutting-edge statistical learning techniques to analyze their data. The text assumes only a previous course in linear regression and no knowledge of matrix algebra.