Set Theory

With an Introduction to Real Point Sets

Author: Abhijit Dasgupta

Publisher: Springer Science & Business Media

ISBN: 1461488540

Category: Mathematics

Page: 444

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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.
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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: 5716

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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.
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The Philosophy of Set Theory

An Historical Introduction to Cantor's Paradise

Author: Mary Tiles

Publisher: Courier Corporation

ISBN: 9780486435206

Category: Mathematics

Page: 239

View: 518

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A century ago, Georg Cantor demonstrated the possibility of a series of transfinite infinite numbers. His methods, unorthodox for the time, enabled him to derive theorems that established a mathematical reality for a hierarchy of infinities. Cantor's innovation was opposed, and ignored, by the establishment; years later, the value of his work was recognized and appreciated as a landmark in mathematical thought, forming the beginning of set theory and the foundation for most of contemporary mathematics. As Cantor's sometime collaborator, David Hilbert, remarked, "No one will drive us from the paradise that Cantor has created." This volume offers a guided tour of modern mathematics' Garden of Eden, beginning with perspectives on the finite universe and classes and Aristotelian logic. Author Mary Tiles further examines permutations, combinations, and infinite cardinalities; numbering the continuum; Cantor's transfinite paradise; axiomatic set theory; logical objects and logical types; and independence results and the universe of sets. She concludes with views of the constructs and reality of mathematical structure. Philosophers with only a basic grounding in mathematics, as well as mathematicians who have taken only an introductory course in philosophy, will find an abundance of intriguing topics in this text, which is appropriate for undergraduate-and graduate-level courses.
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Set Theory

Author: Felix Hausdorff

Publisher: American Mathematical Soc.

ISBN: 9780821838358

Category: Mathematics

Page: 352

View: 4929

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In the early twentieth century, Hausdorff developed an axiomatic approach to topology, which continues to be the foundation of modern topology. The present book, the English translation of the third edition of Hausdorff's Mengenlehre, is a thorough introduction to his theory of point-set topology. The treatment begins with topics in the foundations of mathematics, including the basics of abstract set theory, sums and products of sets, cardinal and ordinal numbers, and Hausdorff's well-ordering theorem. The exposition then specializes to point sets, where major topics such as Borel systems, first and second category, and connectedness are considered in detail. Next, mappings between spaces are introduced. This leads naturally to a discussion of topological spaces and continuous mappings between them. Finally, the theory is applied to the study of real functions and their properties. The book does not presuppose any mathematical knowledge beyond calculus, but it does require a certain maturity in abstract reasoning; qualified college seniors and first-year graduate students should have no difficulty in making the material their own.
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Introduction To Set Theory

Author: A. K. Sharma

Publisher: Discovery Publishing House

ISBN: 9788171418770

Category: Set theory

Page: 118

View: 9569

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This book Introduction to Set Theory is very important in the field of modern algebra. It is very important to study this book to study modern mathematics. This book contain preliminary Notation, Sets, Subsets, Mapping Function and Relation. This book is useful to the students of under graduate, post graduate students and the candidate appearing in various competitions like pre Engineering/I.A.S/ P.C.S. etc. Contents: Preliminary Notation, Relations, Product or Composite of Mapping, Mapping or Functions
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Notes on Set Theory

Author: Yiannis N. Moschovakis

Publisher: Springer

ISBN: 0387941800

Category: Mathematics

Page: 272

View: 3642

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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. At the same time, it is often viewed as a foundation of mathematics so that in the most prevalent, current mathematical practice "to make a notion precise" simply means "to define it in set theory." This book tries to do justice to both aspects of the subject: it gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets (including the basic results that have applications to computer science), but it also attempts to explain precisely how mathematical objects can be faithfully modeled within the universe of sets." "Topics covered include the naive theory of equinumerosity; paradoxes and axioms; modeling mathematical notions by sets; cardinal numbers; natural numbers; fixed points (continuous least-fixed-point theorem); well-ordered sets (transfinite induction and recursion, Hartogs' theorem, comparability of well-ordered sets, least-fixed-point theorem); the Axiom of Choice and its consequences; Baire space (Cantor-Bendixson theorem, analytic pointsets, perfect set theorem); Replacement and other axioms; ordinal numbers. There is an Appendix on the real numbers and another on natural models, including the antifounded universe." "The book is aimed at advanced undergraduate or beginning graduate mathematics students and at mathematically minded graduate students of computer science and philosophy."--BOOK JACKET.
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An Introduction to Random Sets

Author: Hung T. Nguyen

Publisher: CRC Press

ISBN: 9781420010619

Category: Mathematics

Page: 272

View: 5635

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The study of random sets is a large and rapidly growing area with connections to many areas of mathematics and applications in widely varying disciplines, from economics and decision theory to biostatistics and image analysis. The drawback to such diversity is that the research reports are scattered throughout the literature, with the result that in science and engineering, and even in the statistics community, the topic is not well known and much of the enormous potential of random sets remains untapped. An Introduction to Random Sets provides a friendly but solid initiation into the theory of random sets. It builds the foundation for studying random set data, which, viewed as imprecise or incomplete observations, are ubiquitous in today's technological society. The author, widely known for his best-selling A First Course in Fuzzy Logic text as well as his pioneering work in random sets, explores motivations, such as coarse data analysis and uncertainty analysis in intelligent systems, for studying random sets as stochastic models. Other topics include random closed sets, related uncertainty measures, the Choquet integral, the convergence of capacity functionals, and the statistical framework for set-valued observations. An abundance of examples and exercises reinforce the concepts discussed. Designed as a textbook for a course at the advanced undergraduate or beginning graduate level, this book will serve equally well for self-study and as a reference for researchers in fields such as statistics, mathematics, engineering, and computer science.
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An Introduction to Relativistic Quantum Field Theory

Author: Silvan S. Schweber

Publisher: Courier Corporation

ISBN: 0486139603

Category: Science

Page: 944

View: 3450

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In a relatively simple presentation that remains close to familiar concepts, this text for upper-level undergraduates and graduate students introduces modern developments of quantum field theory. "Combines thorough knowledge with a high degree of didactic ability and a delightful style." — Mathematical Reviews. 1961 edition.
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Classic Set Theory

For Guided Independent Study

Author: D.C. Goldrei

Publisher: CRC Press

ISBN: 9780412606106

Category: Mathematics

Page: 296

View: 402

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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 numbers Defining natural numbers in terms of sets The potential paradoxes in set theory The Zermelo-Fraenkel axioms for set theory The axiom of choice The arithmetic of ordered sets Cantor'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.
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Introduction to Complex Analysis

Functions of Several Variables

Author: Boris Vladimirovich Shabat

Publisher: American Mathematical Soc.

ISBN: 0821819755

Category: Functions of complex variables

Page: 371

View: 4292

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Since the 1960s, there has been a flowering in higher-dimensional complex analysis. Both classical and new results in this area have found numerous applications in analysis, differential and algebraic geometry, and, in particular, contemporary mathematical physics. In many areas of modern mathematics, the mastery of the foundations of higher-dimensional complex analysis has become necessary for any specialist. Intended as a first study of higher-dimensional complex analysis, this book covers the theory of holomorphic functions of several complex variables, holomorphic mappings, and submanifolds of complex Euclidean space.
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