Set theory Books

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 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 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 SynopsisAccessible to all students with a sound background in high school mathematics, A Concise Introduction to Pure Mathematics, Fourth Edition presents some of the most fundamental and beautiful ideas in pure mathematics. It covers not only standard material but also many interesting topics not usually encountered at this level, such as the theory of solving cubic equations; Eulerâs formula for the numbers of corners, edges, and faces of a solid object and the five Platonic solids; the use of prime numbers to encode and decode secret information; the theory of how to compare the sizes of two infinite sets; and the rigorous theory of limits and continuous functions.New to the Fourth Edition Two new chapters that serve as an introduction to abstract algebra via the theory of groups, covering abstract reasoning as well as many examples and applications New material on inequalities, counting methods, the inclusion-exclusion princiTrade Review"In addition to preparing students to go on into mathematics, it is also a wonderful choice for a student who will not necessarily go on in mathematics but wants a gentle but fascinating introduction into the culture of mathematics. … This book will give a student the understanding to go on in further courses in abstract algebra and analysis. The notion of a proof will no longer be foreign, but also mathematics will not be viewed as some abstract black box. At the very least, the student will have an appreciation of mathematics. As usual, Liebeck’s writing style is clear and easy to read. This is a book that could be read by a student on his or her own. … While there is a difference in mathematical education between the U.K. and the U.S., this book will serve both groups of students extremely well."—From the Foreword by Robert Guralnick, University of Southern California, Los Angeles, USA "Liebeck’s book stands out from the crowd of similar books by being short (as the title says, it is concise) and by trying to expose students to mathematical ideas beyond the basics of sets and logic. In addition to the pre-Analysis and pre-Algebra chapters, there are chapters on complex numbers, inequalities, some number theory and combinatorics, and the Platonic solids. Students are taught how to understand and create proofs, but they are also given a glimpse of what it is all for…applied mathematicians also need to know about proofs, counting, inequalities, bounds, and even groups — and this book could help them learn all that."—Fernando Q. Gouvêa, MAA Reviews, June 2016 Praise for Previous Editions:" It would in fact be difficult to find in this excellent book three consecutive pages that do not contain material useful to students or practitioners. … A diligent, active reader of this outstanding book will have the best foundation at minimum cost for making meaningful contributions to mathematics, science, or engineering."—Computing Reviews, November 2011"Now in an updated and expanded third edition, A Concise Introduction to Pure Mathematics provides an informed and informative presentation into a representative selection of fundamental ideas in mathematics … . Of special note is the inclusion of solutions to all of the odd-numbered exercises. An ideal, accessible, elegant, student-friendly, and highly recommended choice for classroom textbooks for high school and college-level mathematics curriculums, A Concise Introduction to Pure Mathematics is further enhanced with a selective bibliography, an index of symbols, and a comprehensive index."—Library Bookwatch, December 2010 "This book displays a unique combination of lightness and rigor, leavened with the right dose of humor. When I used it for a course, students could not get enough, and I have been recommending independent study from it to students wishing to take a core course in analysis without having taken the prerequisite course. The material is very well chosen and arranged, and teaching from Liebeck’s book has in many different ways been among my most rewarding teaching experiences during the last decades."—Boris Hasselblatt, Tufts University, Medford, Massachusetts, USA "The book will continue to serve well as a transitional course to rigorous mathematics and as an introduction to the mathematical world … ."—Gerald A. Heuer, Zentralblatt MATH, 2009 "… a pleasure to read … a very welcome and highly accessible book."—Michael Ward, The Mathematical Gazette, March 2007 "In addition to preparing students to go on into mathematics, it is also a wonderful choice for a student who will not necessarily go on in mathematics but wants a gentle but fascinating introduction into the culture of mathematics. … This book will give a student the understanding to go on in further courses in abstract algebra and analysis. The notion of a proof will no longer be foreign, but also mathematics will not be viewed as some abstract black box. At the very least, the student will have an appreciation of mathematics. As usual, Liebeck’s writing style is clear and easy to read. This is a book that could be read by a student on his or her own. … While there is a difference in mathematical education between the U.K. and the U.S., this book will serve both groups of students extremely well."—From the Foreword by Robert Guralnick, University of Southern California, Los Angeles, USA "Liebeck’s book stands out from the crowd of similar books by being short (as the title says, it is concise) and by trying to expose students to mathematical ideas beyond the basics of sets and logic. In addition to the pre-Analysis and pre-Algebra chapters, there are chapters on complex numbers, inequalities, some number theory and combinatorics, and the Platonic solids. Students are taught how to understand and create proofs, but they are also given a glimpse of what it is all for…applied mathematicians also need to know about proofs, counting, inequalities, bounds, and even groups — and this book could help them learn all that."—Fernando Q. Gouvêa, MAA Reviews Praise for Previous Editions:" It would in fact be difficult to find in this excellent book three consecutive pages that do not contain material useful to students or practitioners. … A diligent, active reader of this outstanding book will have the best foundation at minimum cost for making meaningful contributions to mathematics, science, or engineering."—Computing Reviews, November 2011"Now in an updated and expanded third edition, A Concise Introduction to Pure Mathematics provides an informed and informative presentation into a representative selection of fundamental ideas in mathematics … . Of special note is the inclusion of solutions to all of the odd-numbered exercises. An ideal, accessible, elegant, student-friendly, and highly recommended choice for classroom textbooks for high school and college-level mathematics curriculums, A Concise Introduction to Pure Mathematics is further enhanced with a selective bibliography, an index of symbols, and a comprehensive index."—Library Bookwatch, December 2010 "This book displays a unique combination of lightness and rigor, leavened with the right dose of humor. When I used it for a course, students could not get enough, and I have been recommending independent study from it to students wishing to take a core course in analysis without having taken the prerequisite course. The material is very well chosen and arranged, and teaching from Liebeck’s book has in many different ways been among my most rewarding teaching experiences during the last decades."—Boris Hasselblatt, Tufts University, Medford, Massachusetts, USA "The book will continue to serve well as a transitional course to rigorous mathematics and as an introduction to the mathematical world … ."—Gerald A. Heuer, Zentralblatt MATH, 2009 "… a pleasure to read … a very welcome and highly accessible book."—Michael Ward, The Mathematical Gazette, March 2007 Table of ContentsSets and Proofs. Number Systems. Decimals. nth Roots and Rational Powers. Inequalities. Complex Numbers. Polynomial Equations. Induction. Euler’s Formula and Platonic Solids. The Integers. Prime Factorization. More on Prime Numbers. Congruence of Integers. More on Congruence. Secret Codes. Counting and Choosing. More on Sets. Equivalence Relations. Functions. Permutations. Infinity. Introduction to Analysis: Bounds. More Analysis: Limits. Yet More Analysis: Continuity. Introduction to Abstract Algebra: Groups. Introduction to Abstract Algebra: More on Groups. Solutions to Odd-Numbered Exercises. Further Reading. Index of Symbols. 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 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 exploration of a notorious mathematical problem is the work of the man who discovered the solution. The award-winning author employs intuitive explanations and detailed proofs in this self-contained treatment. 1966 edition. Copyright renewed 1994.

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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 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 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 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 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 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 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 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 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 SynopsisIn this book, first published in 2003, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra. Starting with intuitive descriptions of mathematically and physically common phenomena, it leads up to a precise specification of the Category of Sets. Suitable for advanced undergraduates and beginning graduate students.Trade Review"...the categorical approach to mathematics has never been presented with greater conviction than it has in this book. The authors show that the use of categories in analyzing the set concept is not only natural, but inevitable." Mathematical Reviews"To learn set theory this way means not having to relearn it later.... Recommended." ChoiceTable of ContentsForeword; 1. Abstract sets and mappings; 2. Sums, monomorphisms and parts; 3. Finite inverse limits; 4. Colimits, epimorphisms and the axiom of choice; 5. Mapping sets and exponentials; 6. Summary of the axioms and an example of variable sets; 7. Consequences and uses of exponentials; 8. More on power sets; 9. Introduction to variable sets; 10. Models of additional variation; Appendices; Bibliography.

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Book SynopsisThis is an introduction to set theory and logic that starts completely from scratch. The text is accompanied by many methodological remarks and explanations. A rigorous axiomatic presentation of Zermelo-Fraenkel set theory is given, demonstrating how the basic concepts of mathematics have apparently been reduced to set theory. This is followed by a presentation of propositional and first-order logic. Concepts and results of recursion theory are explained in intuitive terms, and the author proves and explains the limitative results of Skolem, Tarski, Church and GÃdel (the celebrated incompleteness theorems). For students of mathematics or philosophy this book provides an excellent introduction to logic and set theory.Trade Review' … written by an excellent mathematician … I very much like the way the author explains things.' European Mathematical SocietyTable of ContentsMathematical induction; 1. Sets and classes; 2. Relations and functions; 3. Cardinals; 4. Ordinals; 5. The axiom of choice; 6. Finite cardinals and alephs; 7. Propositional logic; 8. First order logic; 9. Facts from recursion theory; 10. Limitative results; Appendix: Skolem's paradox.

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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 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 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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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisThe philosophy of mathematics is an exciting subject. Philosophy of Mathematics: Classic and Contemporary Studies explores the foundations of mathematical thought. The aim of this book is to encourage young mathematicians to think about the philosophical issues behind fundamental concepts and about different views on mathematical objects and mathematical knowledge.With this new approach, the author rekindles an interest in philosophical subjects surrounding the foundations of mathematics. He offers the mathematical motivations behind the topics under debate. He introduces various philosophical positions ranging from the classic views to more contemporary ones, including subjects which are more engaged with mathematical logic.Most books on philosophy of mathematics have little to no focus on the effects of philosophical views on mathematical practice, and no concern on giving crucial mathematical results and their philosophical relevance, conTrade Review"The philosophically minded mathematician will find every penny and every second engaged with this book well spent." - Firdous Ahmad Mala, The Mathematical IntelligencerTable of Contents1. Introduction 2. Mathematical Preliminaries 3. Platonism 4. Intuitionism 5. Logicism 6. Formalism 7. Gödel’s Incompleteness Theorem and Computability 8. The Church-Turing Thesis 9. Infinity 10. Supertasks 11. Models, Completeness, and Skolem’s Paradox 12. Axiom of Choice 13. Naturalism 14. Structuralism 15. Yablo’s Paradox 16. Mathematical Pluralism 17. Does Mathematics Need More Axioms? 18. Mathematical Nominalism

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The Baseball Mysteries: Challenging Puzzles for Logical Detectives is a book of baseball puzzles, logical baseball puzzles. To jump in, all you need is logic and a casual fan's knowledge of the game. The puzzles are solved by reasoning from the rules of the game and a few facts.The logic in the puzzles is like legal reasoning. A solution must argue from evidence (the facts) and law (the rules). Unlike legal arguments, however, a solution must reach an unassailable conclusion.There are many puzzle books. But there's nothing remotely like this book. The puzzles here, while rigorously deductive, are firmly attached to actual events, to struggles that are reported in the papers every day.The puzzles offer a unique and scintillating connection between abstract logic and gritty reality.Actually, this book offers the reader an unlimited number of puzzles. Once you've solved a few of the challenges here, every boxscore you see in the pap

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