Notes on Set Theory

Author: Yiannis Moschovakis
Publisher: Springer Science & Business Media
ISBN: 9781475741537
Release Date: 2013-04-17
Genre: Mathematics

What this book is about. The theory of sets is a vibrant, exciting math ematical theory, with its own basic notions, fundamental results and deep open problems, and with significant applications to other mathematical theories. At the same time, axiomatic set theory is often viewed as a foun dation ofmathematics: it is alleged that all mathematical objects are sets, and their properties can be derived from the relatively few and elegant axioms about sets. Nothing so simple-minded can be quite true, but there is little doubt that in standard, current mathematical practice, "making a notion precise" is essentially synonymous with "defining it in set theory. " Set theory is the official language of mathematics, just as mathematics is the official language of science. Like most authors of elementary, introductory books about sets, I have tried to do justice to both aspects of the subject. From straight set theory, these Notes cover the basic facts about "ab stract sets," including the Axiom of Choice, transfinite recursion, and car dinal and ordinal numbers. Somewhat less common is the inclusion of a chapter on "pointsets" which focuses on results of interest to analysts and introduces the reader to the Continuum Problem, central to set theory from the very beginning.

Notes on Logic and Set Theory

Author: P. T. Johnstone
Publisher: Cambridge University Press
ISBN: 0521336929
Release Date: 1987-10-08
Genre: Mathematics

A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included numerous exercises designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics.

Algebraic Set Theory

Author: Andri Joyal
Publisher: Cambridge University Press
ISBN: 0521558301
Release Date: 1995-09-14
Genre: Mathematics

This book offers a new algebraic approach to set theory. The authors introduce a particular kind of algebra, the Zermelo-Fraenkel algebras, which arise from the familiar axioms of Zermelo-Fraenkel set theory. Furthermore, the authors explicitly construct these algebras using the theory of bisimulations. Their approach is completely constructive, and contains both intuitionistic set theory and topos theory. In particular it provides a uniform description of various constructions of the cumulative hierarchy of sets in forcing models, sheaf models and realizability models. Graduate students and researchers in mathematical logic, category theory and computer science should find this book of great interest, and it should be accessible to anyone with a background in categorical logic.

Set Theory

Author: Daniel W. Cunningham
Publisher: Cambridge University Press
ISBN: 9781107120327
Release Date: 2016-07-18
Genre: Mathematics

Set theory can be considered a unifying theory for mathematics. This book covers the fundamentals of the subject.

Elementary Set Theory Part I II

Author: K.T. Leung
Publisher: Hong Kong University Press
ISBN: 9622090265
Release Date: 1992-07-01
Genre: Mathematics

This book provides students of mathematics with the minimum amount of knowledge in logic and set theory needed for a profitable continuation of their studies. There is a chapter on statement calculus, followed by eight chapters on set theory.

Set Theory

Author: Thomas Jech
Publisher: Springer Science & Business Media
ISBN: 9783540447610
Release Date: 2007-05-23
Genre: Mathematics

This monograph covers the recent major advances in various areas of set theory. From the reviews: "One of the classical textbooks and reference books in set theory....The present ‘Third Millennium’ a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master....This well-written book promises to influence the next generation of set theorists, much as its predecessor has done." --MATHEMATICAL REVIEWS

Basic Set Theory

Author: Nikolai Konstantinovich Vereshchagin
Publisher: American Mathematical Soc.
ISBN: 9780821827314
Release Date: 2002
Genre: Mathematics

The main notions of set theory (cardinals, ordinals, transfinite induction) are fundamental to all mathematicians, not only to those who specialize in mathematical logic or set-theoretic topology. Basic set theory is generally given a brief overview in courses on analysis, algebra, or topology, even though it is sufficiently important, interesting, and simple to merit its own dedicated treatment. This book provides just that in the form of a leisurely exposition for a diversified audience. It is suitable for a broad range of readers, from undergraduate students to professional mathematicians who want to finally find out what transfinite induction is and why it is always replaced by Zorn's Lemma. The text introduces all main subjects of ``naive'' (nonaxiomatic) set theory: functions, cardinalities, ordered and well-ordered sets, transfinite induction and its applications, ordinals, and operations on ordinals. Included are discussions and proofs of the Cantor-Bernstein Theorem, Cantor's diagonal method, Zorn's Lemma, Zermelo's Theorem, and Hamel bases. With over 150 problems, the book is a complete and accessible introduction to the subject.

A Book of Set Theory

Author: Charles C Pinter
Publisher: Courier Corporation
ISBN: 9780486497082
Release Date: 2014-07-23
Genre: Mathematics

"This accessible approach to set theory for upper-level undergraduates poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. A historical introduction is followed by discussions of classes and sets, functions, natural and cardinal numbers, the arithmetic of ordinal numbers, and related topics. 1971 edition with new material by the author"--

Introduction to Modern Set Theory

Author: Judith Roitman
Publisher: John Wiley & Sons
ISBN: 0471635197
Release Date: 1990-01-16
Genre: Mathematics

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.

The Joy of Sets

Author: Keith Devlin
Publisher: Springer Science & Business Media
ISBN: 0387940944
Release Date: 1994-06-24
Genre: Mathematics

This text covers the parts of contemporary set theory relevant to other areas of pure mathematics. After a review of "naïve" set theory, it develops the Zermelo-Fraenkel axioms of the theory before discussing the ordinal and cardinal numbers. It then delves into contemporary set theory, covering such topics as the Borel hierarchy and Lebesgue measure. A final chapter presents an alternative conception of set theory useful in computer science.

Set Theory and its Philosophy

Author: Michael Potter
Publisher: Clarendon Press
ISBN: 9780191556432
Release Date: 2004-01-15
Genre: Philosophy

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.

From the Calculus to Set Theory 1630 1910

Author: I. Grattan-Guinness
Publisher: Princeton University Press
ISBN: 0691070822
Release Date: 2000
Genre: Mathematics

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.