Set theory Books

Book SynopsisProofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text''s third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed ''scratch work'' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this booTrade Review'Not only does this book help students learn how to prove results, it highlights why we care so much. It starts in the introduction with some simple conjectures and gathering data, quickly disproving the first but amassing support for the second. Will that pattern persist? How can these observations lead us to a proof? The book is engagingly written, and covers - in clear and great detail - many proof techniques. There is a wealth of good exercises at various levels. I've taught problem solving before (at The Ohio State University and Williams College), and this book has been a great addition to the resources I recommend to my students.' Steven J. Miller, Williams College, Massachusetts'This book is my go-to resource for students struggling with how to write mathematical proofs. Beyond its plentiful examples, Velleman clearly lays out the techniques and principles so often glossed over in other texts.' Rafael Frongillo, University of Colorado, Boulder'I've been using this book religiously for the last eight years. It builds a strong foundation in proof writing and creates the axiomatic framework for future higher-level mathematics courses. Even when teaching more advanced courses, I recommend students to read chapter 3 (Proofs) since it is, in my opinion, the best written exposition of proof writing techniques and strategies. This third edition brings a new chapter (Number Theory), which gives the instructor a few more topics to choose from when teaching a fundamental course in mathematics. I will keep using it and recommending it to everyone, professors and students alike.' Mihai Bailesteanu, Central Connecticut State University'Professor Velleman sets himself the difficult task of bridging the gap between algorithmic and proof-based mathematics. By focusing on the basic ideas, he succeeded admirably. Many similar books are available, but none are more treasured by beginning students. In the Third Edition, the constant pursuit of excellence is further reinforced.' Taje Ramsamujh, Florida International University'Proofs are central to mathematical development. They are the tools used by mathematicians to establish and communicate their results. The developing mathematician often learns what constitutes a proof and how to present it by osmosis. How to Prove It aims at changing that. It offers a systematic introduction to the development, structuring, and presentation of logical mathematical arguments, i.e. proofs. The approach is based on the language of first-order logic and supported by proof techniques in the style of natural deduction. The art of proving is exercised with naive set theory and elementary number theory throughout the book. As such, it will prove invaluable to first-year undergraduate students in mathematics and computer science.' Marcelo Fiore, University of Cambridge'Overall, this is an engagingly-written and effective book for illuminating thinking about and building a careful foundation in proof techniques. I could see it working in an introduction to proof course or a course introducing discrete mathematics topics alongside proof techniques. As a self-study guide, I could see it working as it so well engages the reader, depending on how able they are to navigate the cultural context in some examples.' Peter Rowlett, LMS Newsletter'Altogether this is an ambitious and largely very successful introduction to the writing of good proofs, laced with many good examples and exercises, and with a pleasantly informal style to make the material attractive and less daunting than the length of the book might suggest. I particularly liked the many discussions of fallacious or incomplete proofs, and the associated challenges to readers to untangle the errors in proofs and to decide for themselves whether a result is true.' Peter Giblin, University of Liverpool, The Mathematical GazetteTable of Contents1. Sentential logic; 2. Quantificational logic; 3. Proofs; 4. Relations; 5. Functions; 6. Mathematical induction; 7. Number theory; 8. Infinite sets.

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Book SynopsisIn fall 2000, the Notre Dame logic community hosted Greg Hjorth, Rodney G. Downey, Zoé Chatzidakis, and Paola D'Aquino as visiting lecturers. Each of them presented a month long series of expository lectures at the graduate level. The articles in this volume are refinements of these excellent lectures.

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Book SynopsisLogicism, as put forward by Bertrand Russell, was predicated on a belief that all of mathematics can be deduced from a very small number of fundamental logical principles. In Logicism Renewed, the author revisits this concept in light of advances in mathematical logic and the need for languages that can be understood by both humans and computers that require distinguishing between the intension and extension of predicates. Using Intensional Type Theory (ITT) the author provides a unified foundation for mathematics and computer science, yielding a much simpler foundation for recursion theory and the semantics of computer programs than that currently provided by category theory.

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Book SynopsisThe best introductory text we have seen. Cosmos. Lucidly and gradually explains sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, Boolean algebras, informal axiomatic set theory, several algebraic theories, and 1st-order theories. Its clarity makes this book excellent for self-study.

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Book SynopsisThis book provides an introduction to axiomatic set theory and descriptive set theory. It is written for the upper level undergraduate or beginning graduate students to help them prepare for advanced study in set theory and mathematical logic as well as other areas of mathematics, such as analysis, topology, and algebra.The book is designed as a flexible and accessible text for a one-semester introductory course in set theory, where the existing alternatives may be more demanding or specialized. Readers will learn the universally accepted basis of the field, with several popular topics added as an option. Pointers to more advanced study are scattered throughout the text.

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Book SynopsisThis Element addresses the categoricity arguments that have been more effective in historical cases that reflect philosophically on internal mathematical matters than in recent questions of pre-theoretic metaphysics.Table of Contents1. Introduction; 2. Dedekind in 'Was sind und was sollen die Zahlen?' (1888); 3. Dedekind in 'Was sind und was sollen die Zahlen?' (1888); 4. Kreisel in 'Informal rigor and incompleteness proofs' (1967) and 'Two notes on the foundations of set theory'(1969); 5. Parsons in 'The uniqueness of the natural numbers' (1990) and 'Mathematical induction' (2008); 6. Parsons in 'The uniqueness of the natural numbers' (1990) and 'Mathematical induction' (2008); 7. Conclusion; References.

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Book SynopsisGeared 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.

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Book SynopsisTable of ContentsThe Foundations of Set Theory. Infinitary Combinatorics. The Well-Founded Sets. Easy Consistency Proofs. Defining Definability. The Constructible Sets. Forcing. Iterated Forcing. Bibliography. Indexes.

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Book SynopsisThis authoritative biography of Kurt Goedel relates the life of this most important logician of our time to the development of the field. Goedel's seminal achievements that changed the perception and foundations of mathematics are explained in the context of his life from the turn of the century Austria to the Institute for Advanced Study in Princeton.Trade ReviewDawson's book remains a starting point for our view into the life and work of the man who gave the world incompleteness. -- The Review of Modern Logic, March 2007Table of Contents1. Der Herr Warum (1920-1924) 2. Intellectual Maturation (1924-29) 3. Excursus: A Capsule History of the Development of Logic to 1928 4. Moment of Impact (1929-31) 5. Dozent in absentia (1932-37) 6. “Jetzt, Mengenlehre” (1937-39) 7. Homecoming and Hegira (1939-40) 8. Years of Transition (1940-46) 9. Philosophy and Cosmology (1946-51) 10. Recognition and Reclusion (1951-61) 11. New Light on the Continuum Problem (1961-68) 12. Withdrawal (1969-78) 13. Aftermath 14. Reflections on Gödel’s Life and Legacy

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Book SynopsisThis classic by one of the 20th century''s most prominent mathematicians offers a concise introduction to set theory. Suitable for advanced undergraduates and graduate students in mathematics, it employs the language and notation of informal mathematics. Topics include the basic concepts of set theory, cardinal numbers, transfinite methods, and a good deal more in 25 brief chapters.

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Book SynopsisTable of ContentsContents Preface List of Symbols Chapter 1 Introduction Baby Set Theory Sets—An Informal View Classes Axiomatic Method Notation Historical Notes Chapter 2 Axioms and Operations Axioms Arbitrary Unions and Intersections Algebra of Sets Epilogue Review Exercises Chapter 3 Relations and Functions Ordered Pairs Relations n-Ary Relations Functions Infinite Cartesian Products Equivalence Relations Ordering Relations Review Exercises Chapter 4 Natural Numbers Inductive Sets Peano's Postulates Recursion on ? Arithmetic Ordering on ? Review Exercises Chapter 5 Construction of the Real Numbers Integers Rational Numbers Real Numbers Summaries Two Chapter 6 Cardinal Numbers and the Axiom of Choice Equinumerosity Finite Sets Cardinal Arithmetic Ordering Cardinal Numbers Axiom of Choice Countable Sets Arithmetic of Infinite Cardinals Continuum Hypothesis Chapter 7 Orderings and Ordinals Partial Orderings Well Orderings Replacement Axioms Epsilon-Images Isomorphisms Ordinal Numbers Debts Paid Rank Chapter 8 Ordinals and Order Types Transfinite Recursion Again Alephs Ordinal Operations Isomorphism Types Arithmetic of Order Types Ordinal Arithmetic Chapter 9 Special Topics Well-Founded Relations Natural Models Cofinality Appendix Notation, Logic, and Proofs Selected References for Further Study List of Axioms Index

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Book SynopsisThe transition from school mathematics to university mathematics is seldom straightforward. Students are faced with a disconnect between the algorithmic and informal attitude to mathematics at school, versus a new emphasis on proof, based on logic, and a more abstract development of general concepts, based on set theory. The authors have many years'' experience of the potential difficulties involved, through teaching first-year undergraduates and researching the ways in which students and mathematicians think. The book explains the motivation behind abstract foundational material based on students'' experiences of school mathematics, and explicitly suggests ways students can make sense of formal ideas.This second edition takes a significant step forward by not only making the transition from intuitive to formal methods, but also by reversing the process- using structure theorems to prove that formal systems have visual and symbolic interpretations that enhance mathematical thinking. This is exemplified by a new chapter on the theory of groups. While the first edition extended counting to infinite cardinal numbers, the second also extends the real numbers rigorously to larger ordered fields. This links intuitive ideas in calculus to the formal epsilon-delta methods of analysis. The approach here is not the conventional one of ''nonstandard analysis'', but a simpler, graphically based treatment which makes the notion of an infinitesimal natural and straightforward. This allows a further vision of the wider world of mathematical thinking in which formal definitions and proof lead to amazing new ways of defining, proving, visualising and symbolising mathematics beyond previous expectations.Trade ReviewThe writing is both rigorous and thorough, and the authors use compact presentations to support their explanations and proofs. Highly recommended. * N. W. Schillow, CHOICE *Table of ContentsI: THE INTUITIVE BACKGROUND; II: THE BEGINNINGS OF FORMALISATION; III: THE DEVELOPMENT OF AXIOMATIC SYSTEMS; IV: USING AXIOMATIC SYSTEMS; V: STRENGTHENING THE FOUNDATIONS

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Book SynopsisDesigned for undergraduate students of set theory, this book presents a modern perspective of the classic work of Georg Cantor and Richard Dedekin and their immediate successors. It aims to give students a grounding to the results of set theory as well as to tackle significant problems that arise from the theory.Table of ContentsINTRODUCTIONOutline of the bookAssumed knowledgeTHE REAL NUMBERSIntroductionDedekind's constructionAlternative constructionsThe rational numbersTHE NATURAL NUMBERSIntroductionThe construction of the natural numbersArithmeticFinite setsTHE ZERMELO-FRAENKEL AXIOMSIntroductionA formal languageAxioms 1 to 3Axioms 4 to 6Axioms 7 to 9CARDINAL (Without the Axiom of Choice)IntroductionComparing SizesBasic properties of ˜ and =Infinite sets without AC-countable setsUncountable sets and cardinal arithmetic without ACORDERED SETSIntroductionLinearly ordered setsOrder arithmeticWell-ordered setsORDINAL NUMBERSIntroductionOrdinal numbersBeginning ordinal arithmeticOrdinal arithmeticThe ÀsSET THEORY WITH THE AXIOM OF CHOICEIntroductionThe well-ordering principleCardinal arithmetic and the axiom of choiceThe continuum hypothesisBIBLIOGRAPHYINDEX

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Book SynopsisThis new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures.Trade Review'Recommended for every academic mathematics collection.' Choice'… an excellent introductory textbook on ordered sets and lattices and it is intended for undergraduate and beginning graduate students in mathematics.' Vaclav Slavic, Zentralblatt für Mathematik'I used Introduction to Lattices and Order as the sole textbook in a one semester course. The students enrolled were a heterogeneous group including modestly prepared undergraduates, well trained graduate students, and a few applications-oriented computer science students … In short, the textbook was a success.' Joel Berman, Australian Mathematical Society Gazette'… a well-written, satisfying, informative, and stimulating account of applications that are of great interest, particularly in computer science and social science … it will surely become a classic.' Mathematical Reviews'Altogether, this is a great book. It would be interesting (and educational) to give a course based on it - almost makes me wish I hadn't retired!' Australian Mathematical Society Gazette'… a valuable source to anyone who needs to use ordered structures in any context.' EMS Newsletter'It can be recommended as a valuable source to anyone who needs to use ordered structures in any context.' European Mathematical Society'The book is written in a very engaging and fluid style. The understanding of the content is aided tremendously by the very large number of beautiful lattice diagrams … The book provides a wonderful and accessible introduction to lattice theory, of equal interest to both computer scientists and mathematicians.' Jonathan Cohen, SIGACT NewsTable of ContentsPreface; Preface to the first edition; 1. Ordered sets; 2. Lattices and complete lattices; 3. Formal concept analysis; 4. Modular, distributive and Boolean lattices; 5. Representation theory: the finite case; 6. Congruences; 7. Complete lattices and Galois connections; 8. CPOs and fixpoint theorems; 9. Domains and information systems; 10. Maximality principles; 11. Representation: the general case; Appendix A. A topological toolkit; Appendix B. Further reading; Notation index; Index.

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Book SynopsisThis book introduces readers to the art of doing mathematical proofs. Proofs are the glue that holds mathematics together. They make connections between math concepts and show why things work the way they do. This book teaches the art of proofs using familiar high-school concepts, such as numbers, polynomials, functions, and trigonometry. It retells math as a story, where the next chapter follows from the previous one.Readers will see how various mathematical concepts are tied and will see that mathematics is not a pile of formulas and facts; rather, it has an orderly and beautiful edifice.The author begins with basic rules of logic and then progresses through the topics already familiar to the students: numbers, inequalities, functions, polynomials, exponents, and trigonometric functions. There are also beautiful proofs for conic sections, sequences, and Fibonacci numbers. Each chapter has exercises for the reader.Reviewer Comments:I find the

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Book SynopsisStarting with the most basic notions, Universal Algebra: Fundamentals and Selected Topics introduces all the key elements needed to read and understand current research in this field. Based on the author's two-semester course, the text prepares students for research work by providing a solid grounding in the fundamental constructions and concepts of universal algebra and by introducing a variety of recent research topics.The first part of the book focuses on core components, including subalgebras, congruences, lattices, direct and subdirect products, isomorphism theorems, a clone of operations, terms, free algebras, Birkhoff's theorem, and standard Maltsev conditions. The second part covers topics that demonstrate the power and breadth of the subject. The author discusses the consequences of Jónsson's lemma, finitely and nonfinitely based algebras, definable principal congruences, and the work of Foster and Pixley on primal and quasiprimal algebras. HeTrade Review… as far as I am concerned, the book under review, by Clifford Bergman, is most welcome: we need more of this sort of thing, both for potential universal algebraists and for people like me: fellow travelers to some degree, or mathematicians who both use and thoroughly adore algebra and its structural qualities, and find themselves growing more appreciative of this architectural elegance as they evolve in their work and studies. … it is clearly written and pleasant to read … the author provides motivation as well as examples and exercises galore. At first glance it looks to me like the exercises are well-structured and should do the job of bringing the student or reader along at a decent pace from ignorance to both an appreciation for the subject and some facility with it. It’s definitely an area worth pursuing for a graduate student with the right disposition.—Michael Berg, MAA Reviews, December 2011… excellently written and is highly recommended to all who are interested in universal algebra.—Mathematical ReviewsTable of ContentsFUNDAMENTALS OF UNIVERSAL ALGEBRA: Algebras. Lattices. The Nuts and Bolts of Universal Algebra. Clones, Terms, and Equational Classes. SELECTED TOPICS: Congruence Distributive Varieties. Arithmetical Varieties. Maltsev Varieties. Finite Algebras and Locally Finite Varieties. Bibliography. Index.

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Book SynopsisMichael 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 PhilosophTrade Reviewa wonderful new book . . . Potter has written the best philosophical introduction to set theory on the market * Timothy Bays, Notre Dame Philosophical Reviews *Table of ContentsI. SETS ; II. NUMBERS ; III. CARDINALS AND ORDINALS ; IV. FURTHER AXIOMS

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Book SynopsisThe author identifies natural, real and imaginary numbers with specified physical properties and relations and challenges the myth that mathematical objects can be defined into existence.Trade Review'This book is written with obvious enthusiasm and a deep, and frequently expressed, conviction of the essential correctness of the view it seeks to promote.' Bob Hale, University of St Andrews. THES'This is what philosophy ought to be: a grand vision combined with original and careful work on the details. It is presented with lucidity and modesty and good humour, and bedazzling technicalities. An admirable book.' David Lewis, Princeton University, Australasian Journal of Philosophy'This is what philosophy ought to be: a grand vision combined with original and careful work on the details. It is presented with lucidity and modesty and good humour, and without bedazzling technicalities. An admirable book.' David Lewis, Princeton University, Australasian Journal of PhilosophyTable of ContentsPart I - Metaphysics contains chapter on: Mathematics and universals; Recurrence Part II - Mathematics contains chapters on: Natural Numbers - Pebbles and Pythagoras; Numbers as properties; Numbers as paradigms; Numbers as relations; Numbering sets Real Numbers - Approximations; Arithmetic and Geometry; Proportions; Ratios; Real Numbers Complex Numbers - Imaginary numbers; Complex proportions Sets - From universals to sets; Sets and Essences; Sets and Consistency Part III - Truth and Existence contains chapters on: The Problem - Functions and arguments; Truth and essence; The Fox paradox Wholes and Parts - Counterparts and accidents; Property-instances; Robinson's merger; States of affairs Anyhow to Something - Categories of being; The second-order Fox; Platonism and necessity.

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Book SynopsisSet theory is concerned with the foundation of mathematics. In the original formulations of set theory, there were paradoxes contained in the idea of the set of all sets. Current standard theory (Zermelo-Fraenkel) avoids these paradoxes by restricting the way sets may be formed by other sets, specifically to disallow the possibility of forming the set of all sets. In the 1930s, Quine proposed a different form of set theory in which the set of all sets - the universal set - is allowed, but other restrictions are placed on these axioms. Since then, the steady interest expressed in these non-standard set theories has been boosted by their relevance to computer science.The second edition still concentrates largely on Quine''s New Foundations, reflecting the author''s belief that this provides the richest and most mysterious of the various systems dealing with set theories with a universal set. Also included is an expanded and completely revised account of the set theories of Church-Oswald and Mitchell, with descriptions of permutation models and extensions that preserve power sets. Dr Foster here presents the reader with a useful and readable introduction for those interested in this topic, and a reference work for those already involved in this area.Trade Review...a lively introductin to the current research on NF' * Maruice Boffa, Modern Logic *Table of Contents1. Introduction ; 2. NF and related systems ; 3. Permutation models ; 4. Church-Oswald models ; 5. Open problems ; 6. Bibliography

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Book SynopsisIntermediate Logic is an ideal text for anyone who has taken a first course in logic and is progressing to further study. It examines logical theory, rather than the applications of logic, and does not assume any specific technical grounding. The author introduces and explains each concept and term, ensuring that readers have a firm foundation for study. He provides a broad, deep understanding of logic by adopting and comparing a variety of different methods and approaches.In the first section, Bostock covers such fundamental notions as truth, validity, entailment, qualification, and decision procedures. Part Two lays out a definitive introduction to four key logical tools or procedures: semantic tableaux, axiomatic proofs, natural deduction, and sequent calculi. The final section opens up new areas of existence and identity, concluding by moveing from orthodox logic to an examination of `free logic''.Intermediate Logic provides an ideal secondary course in logic for university studentTrade ReviewThis textbook covers the fundamental proof-theoretical and model-theoretical aspects of classical propositional and first-order logic. . . .The book is clearly written and ideally suited for an intermediate course on the subject, requiring just some elementary knowledge of proof theory and model theory. * Mathematical Reviews *

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Book SynopsisThis third edition, now available in paperback, is a follow up to the author''s classic Boolean-Valued Models and Independence Proofs in Set Theory,. It provides an exposition of some of the most important results in set theory obtained in the 20th century: the independence of the continuum hypothesis and the axiom of choice. Aimed at graduate students and researchers in mathematics, mathematical logic, philosophy, and computer science, the third edition has been extensively updated with expanded introductory material, new chapters, and a new appendix on category theory. It covers recent developments in the field and contains numerous exercises, along with updated and increased coverage of the background material. This new paperback edition includes additional corrections and, for the first time, will make this landmark text accessible to students in logic and set theory.Trade ReviewBell's presentation is lively and pleasent to read, and the material is given in a nicely cohesive way. * Philosophia Mathmatica *Table of ContentsFORWARD BY DANA SCOTT; PREFACE; LIST OF PROBLEMS; APPENDIX: BOOLEAN AND HEYTING ALGEBRA-VALUED MODELS AS CATEGORIES; HISTORICAL NOTES; BIBLIOGRAPHY; INDEX OF SYMBOLS; INDEX OF TERMS

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Book SynopsisMathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. The axioms of set theory have long played this role, so the question of how they are properly judged is of central importance. Maddy discusses the appropriate methods for such evaluations and the philosophical backdrop that makes them appropriate.Trade Review'an engaging contribution to an important philosophical debate [which] deserves to be read far beyond the ranks of philosophers of mathematics' * Journal of Philosophy *Table of ContentsIntroduction ; 1. The Problem ; 2. Proper Method ; 3. Thin Realism ; 4. Arealism ; 5. Morals ; Bibliography

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Book SynopsisDiscrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover''s search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.Table of ContentsPreface; 1. Grover search; 2. Two reflections; 3. Applications; 4. Averaging: 5. Covers and embeddings; 6. Vertex-face walks; 7. Shunts; 8. 1-Dimensional walks; References; Glossary; Index.

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Book SynopsisThis book is designed for readers who knowelementary mathematical logic and axiomatic settheory, and who want to learn more about set theory.The primary focus of the book is on the independenceproofs. Most famous among these is the independenceof the Continuum Hypothesis (CH); that is, there aremodels of the axioms of set theory (ZFC) in whichCH is true, and other models in which CH is false.More generally, cardinal exponentiation on the regularcardinals can consistently be anything not contradictingthe classical theorems of Cantor and König.The basic methods for the independence proofs arethe notion of constructibility, introduced by Gödel, andthe method of forcing, introduced by Cohen. This bookdescribes these methods in detail, verifi es the basicindependence results for cardinal exponentiation, andalso applies these methods to prove the independenceof various mathematical questions in measure theoryand general topology.Before the chapters on forcing, there is a fairly longchapter on "infi nitary combinatorics". This consistsof just mathematical theorems (not independenceresults), but it stresses the areas of mathematicswhere set-theoretic topics (such as cardinal arithmetic)are relevant.There is, in fact, an interplay between infi nitarycombinatorics and independence proofs. Infi nitarycombinatorics suggests many set-theoretic questionsthat turn out to be independent of ZFC, but it alsoprovides the basic tools used in forcing arguments. Inparticular, Martin''s Axiom, which is one of the topicsunder infi nitary combinatorics, introduces many of thebasic ingredients of forcing.

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Book Synopsis1 Fundamental of Crisp Set Theory.- 2 Fundamental Concepts of Fuzzy Sets.- 3 Generalization of Fuzzy Sets.- 4 Decomposition of a Fuzzy Set and Extension Principle.- 5 Fuzzy Set-Theoretic Operators.- 6 Arithmetic Operations and Fuzzy Mathematics.- 7 Fuzzy Relations.

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Book SynopsisIntroduction.- Functions and Relations.- Ordinal and Cardinal Numbers.- Applications in Other Branches of Mathematics.- Banach-Tarski Paradox.

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Book SynopsisInformation granules, as encountered in natural language, are implicit in nature. To make them fully operational so they can be effectively used to analyze and design intelligent systems, information granules need to be made explicit. An emerging discipline, granular computing focuses on formalizing information granules and unifying them to create a coherent methodological and developmental environment for intelligent system design and analysis. Granular Computing: Analysis and Design of Intelligent Systems presents the unified principles of granular computing along with its comprehensive algorithmic framework and design practices. Introduces the concepts of information granules, information granularity, and granular computing Presents the key formalisms of information granules Builds on the concepts of information granules with discussion of higher-order and higher-type information granules Discusses the operational concept of inTrade Review"Dr. Pedrycz is an internationally acclaimed authority in the granular computing area. ... I particularly appreciate his elegant writing style. This book is the first comprehensive treatise of the granular computing techniques and their application to the design of intelligent systems. ... As an application-oriented practitioner in computational intelligence systems, I think that this book will be a welcome and strongly needed addition to this field. I cannot think of any other expert worldwide more qualified than Prof. Pedycz to write such a book."—Emil M. Petriu, University of Ottawa, Canada "This volume covers most of the interesting and important topics in granular computing. The contents may be well understood by senior or master course students in the field of computer science ... also a good textbook for engineers who are involved in developing so-called intelligent systems."—Kaoru Hirota, Tokyo Institute of Technology, Japan "Dr. Pedrycz’s latest magnum opus ... breaks new ground in many directions. [It] takes an important step toward achievement of human-level machine intelligence—a principal goal of artificial intelligence (AI) since its inception. ... [This is] a remarkably well put together and reader-friendly collection of concepts and techniques, which constitute granular computing. ... [The book] combines extraordinary breadth with extraordinary depth. It contains a wealth of new ideas, and unfolds a vast panorama of concepts, methods, and applications. ... Dr. Pedrycz’s development and description of these concepts, techniques, and their applications is a truly remarkable achievement. ... must reading for all who are concerned with the design and application of intelligent systems."—From the Foreword by Lotfi A. Zadeh, University of California, Berkeley, USA Table of ContentsInformation Granularity, Information Granules, and Granular Computing. Key Formalisms for Representation of Information Granules and Processing Mechanisms. Information Granules of Higher Type and Higher Order, and Hybrid Information Granules. Representation of Information Granules. The Design of Information Granules. Optimal Allocation of Information Granularity: Building Granular Mappings. Granular Description of Data and Pattern Classification. Granular Models: Architectures and Development. Granular Time Series. From Models to Granular Models. Collaborative and Linguistic Models of Decision Making. Index.

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Book SynopsisSet Theoretical Aspects of Real Analysis is built around a number of questions in real analysis and classical measure theory, which are of a set theoretic flavor. Accessible to graduate students, and researchers the beginning of the book presents introductory topics on real analysis and Lebesgue measure theory. These topics highlight the boundary between fundamental concepts of measurability and nonmeasurability for point sets and functions. The remainder of the book deals with more specialized material on set theoretical real analysis. The book focuses on certain logical and set theoretical aspects of real analysis. It is expected that the first eleven chapters can be used in a course on Lebesque measure theory that highlights the fundamental concepts of measurability and non-measurability for point sets and functions. Provided in the book are problems of varying difficulty that range from simple observations to advanced results. Relatively difficult Table of ContentsZF theory and some point sets on the real line. Countable versions of AC and real analysis. Uncountable versions of AC and Lebesgue nonmeasurable sets. The Continuum Hypothesis and Lebesgue nonmeasurable sets. Measurability properties of sets and functions. Radon measures and nonmeasurable sets. Real-valued step functions with strange measurability properties. Relationships between certain classical constructions of Lebesgue nonmeasurable sets. Measurability properties of Vitali sets. A relationship between the measurability and continuity of real-valued functions. A relationship between absolutely nonmeasurable functions and Sierpinski-Zygmund functions. Sums of absolutely nonmeasurable injective functions. A large group of absolutely nonmeasurable additive functions. Additive properties of certain classes of pathological functions. Absolutely nonmeasurable homomorphisms of commutative groups. Measurable and nonmeasurable sets with homogeneous sections. A combinatorial problem on translation invariant extensions of the Lebesgue measure. Countable almost invariant partitions of G-spaces. Nonmeasurable unions of measure zero sections of plane sets. Measurability properties of well-orderings. Appendices. Bibliography. Subject Index.

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Book SynopsisA Bridge to Higher Mathematics is more than simply another book to aid the transition to advanced mathematics. The authors intend to assist students in developing a deeper understanding of mathematics and mathematical thought. The only way to understand mathematics is by doing mathematics. The reader will learn the language of axioms and theorems and will write convincing and cogent proofs using quantifiers. Students will solve many puzzles and encounter some mysteries and challenging problems. The emphasis is on proof. To progress towards mathematical maturity, it is necessary to be trained in two aspects: the ability to read and understand a proof and the ability to write a proof. The journey begins with elements of logic and techniques of proof, then with elementary set theory, relations and functions. Peano axioms for positive integers and for natural numbers follow, in particular mathematical and other forms of induction. Next Trade ReviewThis is one of the shorter books for a course that introduces students to the concept of mathematical proofs. The brevity is due to the "bare-bones" nature of the treatment. The number of topics covered, the number of examples, and the number of exercises are not smaller than what appears in competing textbooks; what is shorter is the text that one finds between theorems, lemmas, examples, and exercises. Besides the topics found in similar textbooks (i.e., proof techniques, logic, set theory, relations, and functions), there are chapters on (very) elementary number theory, combinatorial counting techniques, and Peano axioms on the set of positive integers. Several chapters are devoted to the construction of various kinds of numbers, such as integers, rationals, real numbers, and complex numbers. Answers to around half the exercises are included at the end of the book, and a few have complete solutions. This reviewer finds the book more enjoyable than the average competing textbook. --M. Bona, University of FloridaTable of ContentsElements of logicTrue and false statementsLogical connectives and truth tablesLogical equivalenceQuantifiersProofs: Structures and strategiesAxioms, theorems and proofsDirect proofContrapositive proofProof by equivalent statementsProof by casesExistence proofsProof by counterexampleProof by mathematical inductionElementary Theory of Sets. FunctionsAxioms for set theoryInclusion of setsUnion and intersection of setsComplement, difference and symmetric difference of setsOrdered pairs and the Cartersian productFunctionsDefinition and examples of functionsDirect image, inverse imageRestriction and extension of a functionOne-to-one and onto functionsComposition and inverse functions*Family of sets and the axiom of choiceRelationsGeneral relations and operations with relationsEquivalence relations and equivalence classesOrder relations*More on ordered sets and Zorn's lemmaAxiomatic theory of positive integersPeano axioms and additionThe natural order relation and subtractionMultiplication and divisibilityNatural numbersOther forms of inductionElementary number theoryAboslute value and divisibility of integersGreatest common divisor and least common multipleIntegers in base 10 and divisibility testsCardinality. Finite sets, infinite setsEquipotent setsFinite and infinite setsCountable and uncountable setsCounting techniques and combinatoricsCounting principlesPigeonhole principle and parityPermutations and combinationsRecursive sequences and recurrence relationsThe construction of integers and rationals Definition of integers and operationsOrder relation on integersDefinition of rationals, operations and orderDecimal representation of rational numbersThe construction of real and complex numbersThe Dedekind cuts approachThe Cauchy sequences approachDecimal representation of real numbersAlgebraic and transcendental numbersComples numbersThe trigonometric form of a complex number

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Book SynopsisHighly technical monograph in which the authors, writing on the basis of their own recent research for the benefit of expert readers, describe a general theory of stochastic integration equations. First published in 1990.Table of ContentsIntroduction, Nonlinear Stochastic Integrators, Stochastic Calculus, Dependence on the initial Conditions and Flows.

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Book SynopsisThis book presents articles of L.V. Kantorovich on the descriptive theory of sets and function and on functional analysis in semi-ordered spaces, to demonstrate the unity of L.V. Kantorovich's creative research. It also includes two papers on the extension of Hilbert space.

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Book SynopsisThere are many situations in science and engineering where complex output data from a given system is used to formulate a model of how that system operates, or to simulate its response to different inputs. Applications include control, decision theory, and the emerging fields of bioinformatics.Trade Review"To cope with real world uncertainties and provide a philosophical and practical guide...several methodologies are presented..." (SciTech Book News, Vol. 25, No. 4, December 2001) "...certainly a book that should be in the library of any institution where research and advanced study in fuzzy systems are carried out." (Choice, Vol. 39, No. 7, March 2002) "...well organized, easy to read, and self-contained.... I would recommend it to anyone interested in self-study of the basic ideas of fuzzy systems..." (International Journal of General Systems, Vol. 31, No. 6, 2002)Table of ContentsPreface. Acknowledgments. Introduction. System Analysis. Uncertainty Techniques. Learning from Data: System Identification. Propositions as Subsets of the Data Space. Fuzzy Systems and Identification. Random-Set Modelling and Identification. Certain Uncertainty. Fuzzy Inference Engines. Fuzzy Classification. Fuzzy Control. Fuzzy Mathematics. Summary. Appendices. Index.

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Book SynopsisFuzzy clustering, which combines fuzzy logic and cluster analysis techniques, has experienced a spur of interest in recent years owing to its important applications in image recognition. This revised, updated, and expanded translation of the German book deals with the ideas and algorithms of fuzzy clustering and their applications.Table of ContentsIntroduction. Basic Concepts. Classical Fuzzy Clustering Algorithms. Linear and Ellipsoidal Prototypes Shell Prototypes. Polygonal Object Boundaries. Cluster Estimation Models. Cluster Validity. Rule Generation with Clustering. Appendix. Bibliography.

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Book SynopsisProvides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. This title delves into arguments originating in Nori's work that have been further developed by others.Trade Review"This dense, fascinating book by Voisin is a report of some of the exciting discoveries she has made in the quest of the secrets of algebraic cycles."--Alberto Collino, Zentralblatt MATH "[An advanced] reader will find a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field. Any complex geometer interested in the interplay between algebraic cycles, Hodge theory and algebraic topology should have this book on his or her shelf."--C. A. M. Peters, Mathematical Reviews ClippingsTable of ContentsPreface vii 1Introduction 1 1.1 Decomposition of the diagonal and spread 3 1.2 The generalized Bloch conjecture 7 1.3 Decomposition of the small diagonal and application to the topology of families 9 1.4 Integral coefficients and birational invariants 11 1.5 Organization of the text 13 2Review of Hodge theory and algebraic cycles 15 2.1 Chow groups 15 2.2 Hodge structures 24 3Decomposition of the diagonal 36 3.1 A general principle 36 3.2 Varieties with small Chow groups 44 4Chow groups of large coniveau complete intersections 55 4.1 Hodge coniveau of complete intersections 55 4.2 Coniveau 2 complete intersections 64 4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67 4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86 5On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88 5.1 Tautological ring of a K3 surface 88 5.2 A decomposition of the small diagonal 96 5.3 Deligne's decomposition theorem for families of K3 surfaces 106 6Integral coefficients 123 6.1 Integral Hodge classes and birational invariants 123 6.2 Rationally connected varieties and the rationality problem 127 6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139 Bibliography 155 Index 163

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Book SynopsisHave you ever played the addictive card game SET? Have you ever wondered about the connections between games and mathematics? If the answer to either question is yes, then The Joy of SET is the book for you! The Joy of SET takes readers on a fascinating journey into this seemingly simple card game and reveals its surprisingly deep and diverse matTrade Review"[A] model of mathematical exposition. The quality of writing is consistently high: clear but not condescending, humorous, chatty, and a genuine pleasure to read... I doubt it will be very long before I find something from [The Joy of SET] to use in one of my classes."--Mark Hunacek, MAA Reviews "[The Joy of SET] shows how budding interest in mathematics can be fostered and developed... If [middle and high school teachers] ever try to enliven their classes or just interaction with curious students, this book is one they may depend on."--Alexander Bogomolny, Cut the Knot blog "[A]mazing... What I love about The Joy of SET is that it is written in such a way that it can be read and enjoyed by both SET enthusiasts and someone that has never played SET before... Really and truly, there is enough math in this book to keep you busy for a lifetime! ... I definitely recommend this book."--Sarah Carter, Math Equals Love blog "[The Joy of SET] takes readers on a fascinating journey into this seemingly simple card game... The book is in my view just the right way to talk about math as fun, and intellectually challenging."--Robert Harington, Scholarly Kitchen "[A]s the authors convincingly demonstrate ... the mathematics behind SET actually goes very deep... [The Joy of SET] would make a fantastic resource for a middle school, high school, or undergraduate math club."--Brent Yorgey, Math Less Traveled blog "This book, written by a mathematically inclined family, is the first and only work to explore the connection between the game and mathematics... [The Joy of SET] will attract those who play SET and those who want to explore mathematically related subjects."--ChoiceTable of ContentsPreface vii 1 SET and You 1 2 Counting Fun! 27 3 Probability! 52 4 SET and Modular Arithmetic 72 5 SET and Geometry 98 Interlude: How to Improve at SET 136 6 More Combinatorics 149 7 Probability and Statistics 171 8 Vectors and Linear Algebra 197 9 Affine Geometry Plus 229 10 Computing and Simulations 256 Conclusion 288 Solutions to Exercises 289 Bibliography 303 Index 307

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