*With an Introduction to Real Point Sets*

**Author**: Abhijit Dasgupta

**Publisher:**Springer Science & Business Media

**ISBN:**1461488540

**Category:**Mathematics

**Page:**444

**View:**7001

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# Search Results for: set-theory-with-an-introduction-to-real-point-sets

*With an Introduction to Real Point Sets*

**Author**: Abhijit Dasgupta

**Publisher:** Springer Science & Business Media

**ISBN:** 1461488540

**Category:** Mathematics

**Page:** 444

**View:** 7001

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.

**Author**: Joseph Breuer

**Publisher:** Courier Corporation

**ISBN:** 0486154874

**Category:** Mathematics

**Page:** 128

**View:** 8822

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.
*An Introduction to Set Theory and Analysis*

**Author**: John Stillwell

**Publisher:** Springer Science & Business Media

**ISBN:** 331901577X

**Category:** Mathematics

**Page:** 244

**View:** 9303

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.

**Author**: Alexander Kechris

**Publisher:** Springer Science & Business Media

**ISBN:** 1461241901

**Category:** Mathematics

**Page:** 404

**View:** 9844

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.
*A Critical Introduction*

**Author**: Michael Potter

**Publisher:** Clarendon Press

**ISBN:** 0191556432

**Category:** Philosophy

**Page:** 360

**View:** 6823

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.

**Author**: Herbert B. Enderton

**Publisher:** Academic Press

**ISBN:** 0080570429

**Category:** Mathematics

**Page:** 279

**View:** 4747

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.

**Author**: Yiannis N. Moschovakis

**Publisher:** Springer Science & Business Media

**ISBN:** 9783540941804

**Category:** Axiomatische Mengenlehre

**Page:** 272

**View:** 8390

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.

**Author**: E. Kamke

**Publisher:** Courier Corporation

**ISBN:** 048645083X

**Category:** Mathematics

**Page:** 144

**View:** 2243

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.

**Author**: Kazimierz Kuratowski

**Publisher:** Elsevier

**ISBN:** 1483151638

**Category:** Mathematics

**Page:** 352

**View:** 5587

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.
*An Introduction with Application to Topological Groups*

**Author**: George McCarty

**Publisher:** Courier Corporation

**ISBN:** 0486450821

**Category:** Mathematics

**Page:** 288

**View:** 7411

This stimulating introduction employs the language of point set topology to define and discuss topological groups. It examines set-theoretic topology and its applications in function spaces as well as homotopy and the fundamental group. Well-chosen exercises and problems serve as reinforcements. 1967 edition. Includes 99 illustrations.

**Author**: Paul J. Cohen,Martin Davis

**Publisher:** Courier Corporation

**ISBN:** 0486469212

**Category:** Mathematics

**Page:** 154

**View:** 5728

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.
*For Guided Independent Study*

**Author**: D.C. Goldrei

**Publisher:** Routledge

**ISBN:** 1351460609

**Category:** Mathematics

**Page:** 296

**View:** 4355

Designed for undergraduate students of set theory, Classic Set Theory presents a modern perspective of the classic work of Georg Cantor and Richard Dedekin and their immediate successors. This includes:The definition of the real numbers in terms of rational numbers and ultimately in terms of natural numbersDefining natural numbers in terms of setsThe potential paradoxes in set theoryThe Zermelo-Fraenkel axioms for set theoryThe axiom of choiceThe arithmetic of ordered setsCantor's two sorts of transfinite number - cardinals and ordinals - and the arithmetic of these.The book is designed for students studying on their own, without access to lecturers and other reading, along the lines of the internationally renowned courses produced by the Open University. There are thus a large number of exercises within the main body of the text designed to help students engage with the subject, many of which have full teaching solutions. In addition, there are a number of exercises without answers so students studying under the guidance of a tutor may be assessed.Classic Set Theory gives students sufficient grounding in a rigorous approach to the revolutionary results of set theory as well as pleasure in being able to tackle significant problems that arise from the theory.
*Platonism and Circularity; Translations of Paul Finsler's Papers on Set Theory with Introductory Comments*

**Author**: Paul Finsler,David Booth

**Publisher:** Springer Science & Business Media

**ISBN:** 9783764354008

**Category:** Mathematics

**Page:** 278

**View:** 6263

This English translation of Paul Finsler's papers on set theory reflects the central concerns of his investigations, namely the philosophical, foundational, and combinatorial approaches. Each section contains an introduction to the field by the editors, and a technical background is not necessary.
*An Introductory History*

**Author**: I. Grattan-Guinness,H. J. M. Bos

**Publisher:** Princeton University Press

**ISBN:** 9780691070827

**Category:** Mathematics

**Page:** 306

**View:** 6482

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.
*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:** 9821

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.

**Author**: Terence Tao

**Publisher:** American Mathematical Soc.

**ISBN:** 0821869191

**Category:** Mathematics

**Page:** 206

**View:** 2580

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.

**Author**: Judith Roitman

**Publisher:** John Wiley & Sons

**ISBN:** 9780471635192

**Category:** Mathematics

**Page:** 156

**View:** 2127

This is modern set theory from the ground up--from partial orderings and well-ordered sets to models, infinite cobinatorics and large cardinals. The approach is unique, providing rigorous treatment of basic set-theoretic methods, while integrating advanced material such as independence results, throughout. The presentation incorporates much interesting historical material and no background in mathematical logic is assumed. Treatment is self-contained, featuring theorem proofs supported by diagrams, examples and exercises. Includes applications of set theory to other branches of mathematics.

**Author**: Patrick Suppes

**Publisher:** Courier Corporation

**ISBN:** 0486136876

**Category:** Mathematics

**Page:** 265

**View:** 9770

Geared toward upper-level undergraduates and graduate students, this treatment examines the basic paradoxes and history of set theory and advanced topics such as relations and functions, equipollence, more. 1960 edition.

**Author**: Krzysztof Ciesielski

**Publisher:** Cambridge University Press

**ISBN:** 9780521594653

**Category:** Mathematics

**Page:** 236

**View:** 1790

Presents those methods of modern set theory most applicable to other areas of pure mathematics.

**Author**: Maxwell Rosenlicht

**Publisher:** Courier Corporation

**ISBN:** 0486134687

**Category:** Mathematics

**Page:** 272

**View:** 4013

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.

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