Mathematical logic Books

Book SynopsisThe Quine-Putnam indispensability argument in the philosophy of mathematics urges us to place mathematical entities on the same ontological footing as other theoretical entities essential to our best scientific theories. Recently, the argument has come under serious scrutiny, with many influential philosophers unconvinced of its cogency. This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.Trade ReviewOverall, the book presents a clear picture of the Quinean world view. * Mathematical Reviews *

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Book SynopsisMathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positionsTrade Review"The Oxford Handbook of the Philosophy of Mathematics and Logic is most certainly here to stay for a very long time. The quality of each of the contributions is reflected in the authors' stimulating writing. The handbook can add substantially to the emerging thoughts and studies on the subject."--Current Engineering Practice"The Oxford Handbook of the Philosophy of Mathematics and Logic is a very accessible, wide ranging work that serves not only to indicate the 'state of the art' in the given area, but, remarkably, also serves as a very fine introduction to the field. I recommend it highly, both to workers in the given field and, equally, to the 'general philosopher,' regardless of one's main area." --Notre Dame Philosophical Reviews

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Book SynopsisStewart Shapiro presents a distinctive and persuasive view of the foundations of mathematics, arguing controversially that second-order logic has a central role to play in laying these foundations. To support this contention, he first gives a detailed development of second-order and higher-order logic, in a way that will be accessible to graduate students. He then demonstrates that second-order notions are prevalent in mathematics as practised, and that higher-order logic is needed to codify many contemporary mathematical concepts. Throughout, he emphasizes philosophical and historical issues that the subject raises. Foundations without Foundationalism is a key contribution both to philosophy of mathematics and to mathematical logic. ''In this excellent treatise Shapiro defends the use of second-order languages and logic as framework for mathematics. His coverage of the wide range of logical and philosophical topics required for understanding the controversy over second-order logic isTrade ReviewContains more on second-order logic than is readily available in any other textbook or survey. Philosophically, the book also contains many words of wisdom. * Journal of Symbolic Logic *Table of ContentsPART I: ORIENTATION; 1. TERMS AND QUESTIONS; 2. FOUNDATIONALISM AND FOUNDATIONS OF MATHEMATICS; PART II: LOGIC AND MATHEMATICS; 3. THEORY; 4. METATHEORY; 5. SECOND-ORDER LOGIC AND MATHEMATICS; 6. ADVANCED METATHEORY; PART III: HISTORY AND PHILOSOPHY; 7. THE HISTORICAL 'TRIUMPH' OF FIRST-ORDER LANGUAGES; 8. SECOND-ORDER LOGIC AND RULE-FOLLOWING; 9. THE COMPETITION; REFERENCES; INDEX

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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 SynopsisThe ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author''s teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course.Trade Review'a clear and unifying treatment of fundamental concepts underlying Computer Sciences and Foundations of Mathematics' Professor Boris Zilber (Professor of Mathematical Logic, University of Oxford)'an excellent book' Professor Dov Gabbay (King's College, London)Table of ContentsPreliminaries ; 1. Propositional Logic ; 2. Structures and First-Order Logic ; 3. Proof Theory ; 4. Properties of First-Order Logic ; 5. First-Order Theories ; 6. Models of Countable Theories ; 7. Computability and Complexity ; 8. The Incompleteness Theorems ; 9. Beyond First-Order Logic ; 10. Finite Model Theory ; Bibliography ; Index

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Book SynopsisTechniques for reasoning about actions an change in the physical world is one of the classical research topics in artificial intelligence. It is motivated by the needs of autonomous robots which must be able to anticipate their immediate future, to plan their future actions, and to figure out what went wrong in case of problems. It is also motivated by the needs of common-sense reasoning for example in the understanding of natural language texts, where processes and change over time is an ever-present phenomenon. The same set of problems arises in several other areas of computing such as in conceptual modelling for data bases, and in the rapidly growing area of intelligent control.The present research monograph presents and uses a novel methodology for reasoning about actions and change. Traditional research contributions have proposed new logic variants which were only supported by episodical examples. THe work described here uses a systematic methodology for identifying the exact ranTrade ReviewThe book presents deep and serious insight into inert and inhabited dynamical systems (IDS). * Zentralblatt fur Mathematik *Those working in nonmonotonic reasoning, planning, temporal logic, reasoning about actions and change, and related areas will find this book worth reading. * Computing Reviews *Table of ContentsInert and inhabited dynamical systems ; Inference operations on scenario descriptions ; Underlying semantics for IDS worlds ; Elementary feature logic and meta-logical concepts ; Lexical-domain object-feature logic ; Temporal feature logic for discrete time domains ; Chronicle completion in k-IA ; Intended models for chronicles in k-IA ; Entailment methods for k-IA using DFL-1 ; Duration constraints ; Entailment methods for k-OA using occlusion ; Composite actions ; Upper applicability bounds and assessment of soundness ; Future directions ; Terms index ; Notation ; References to related work

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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 SynopsisHow strongly should you believe the various propositions that you can express?That is the key question facing Bayesian epistemology. Subjective Bayesians hold that it is largely (though not entirely) up to the agent as to which degrees of belief to adopt. Objective Bayesians, on the other hand, maintain that appropriate degrees of belief are largely (though not entirely) determined by the agent''s evidence. This book states and defends a version of objective Bayesian epistemology. According to this version, objective Bayesianism is characterized by three norms: Probability - degrees of belief should be probabilities Calibration - they should be calibrated with evidence Equivocation - they should otherwise equivocate between basic outcomesObjective Bayesianism has been challenged on a number of different fronts. For example, some claim it is poorly motivated, or fails to handle qualitative evidence, or yields counter-intuitive degrees of belief after updating, or suffers from a failureTable of ContentsPreface ; 1. Introduction ; 2. Objective Bayesianism ; 3. Motivation ; 4. Updating ; 5. Predicate Languages ; 6. Objective Bayesian Nets ; 7. Probabilistic Logic ; 8. Judgement Aggregation ; 9. Languages and Relativity ; 10. Objective Bayesianism in Perspective ; References ; Index

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Book SynopsisBob Hale and Crispin Wright draw together here the key writings in which they have worked out their distinctive neo-Fregean approach to the philosophy of mathematics. The two main components in Frege''s mathematical philosophy were his platonism and his logicism -- the claims, respectively, that mathematics is a body of knowledge about independently existing objects, and that this knowledge may be acquired on the basis of general logical laws and suitable definitions. The central thesis of this collection is that Frege was -- his own eventual recantation notwithstanding -- substantially right in both claims. Where neo-Fregeanism principally differs from Frege is in taking a more optimistic view of the kind of contextual explanation (proceeding via what are now commonly called abstraction principles) of the fundamental concepts of arithmetic and analysis which Frege considered and rejected. On this basis, neo-Fregeanism promises defensible and attractive answers to some of the most impoTable of ContentsI. ONTOLOGY AND ABSTRACTION PRINCIPLES ; II. RESPONSES TO CRITICS ; III. HUME'S PRINCIPLE ; IV. ON THE DIFFERENTIATION OF ABSTRACTA ; V. BEYOND NUMBER-THEORY

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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 SynopsisPacked with more than a hundred color illustrations and a wide variety of puzzles and brainteasers, Taking Sudoku 2eriously uses this popular craze as the starting point for a fun-filled introduction to higher mathematics. How many Sudoku solution squares are there? What shapes other than three-by-three blocks can serve as acceptable Sudoku regions? What is the fewest number of starting clues a sound Sudoku puzzle can have? Does solving Sudoku require mathematics? Jason Rosenhouse and Laura Taalman show that answering these questions opens the door to a wealth of interesting mathematics. Indeed, they show that Sudoku puzzles and their variants are a gateway into mathematical thinking generally. Among many topics, the authors look at the notion of a Latin square--an object of long-standing interest to mathematicians--of which Sudoku squares are a special case; discuss how one finds interesting Sudoku puzzles; explore the connections between Sudoku, graph theory, and polynomials; and cTrade ReviewThis well-written book would be of interest to anyone, mathematician or not, who likes solving Sudoku puzzles. * Donald Keedwell, Mathematical Gazette *This is an interesting book. The style is conversational and east to read ... * John Sykes, Mathematics in School *I thoroughly enjoyed this book and do not have any criticisms to make. The authors have produced a lovely addition to any budding or practiced mathematicians bookcase. Well-presented and readable for both the novice and the maths expert, which is an admirable feat, this book is for anyone with an interest, no matter how vague or intense, in Sudoku. * Angie Wade, Significance *A beautiful book. * Paul Levrie, Karel de Grote University College *Table of Contents1. Playing the Game ; Mathematics as Applied Puzzle-Solving ; 2. Latin Squares ; What Do Mathematicians Do? ; 3. Greco-Latin Squares ; The Problem of the Thirty-Six Officers ; 4. Counting ; It's Harder Than it Looks ; 5. Equivalence Classes ; The Importance of Being Essentially Identical ; 6. Searching ; The Art of Finding Needles in Haystacks ; 7. Graphs ; Dots, Lines and Sudoku ; 8. Polynomials ; We Finally Found a Use For Algebra ; 9. Extremes ; Sudoku Pushed to its Limits ; 10. Epilogue ; You Can Never Have Too Many Puzzles ; Solutions to Puzzles

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Book SynopsisThis is the fascinating story of two women who lives were guided by a passion for mathematics and an insatiable curiosity to know and understand the world around them -- the beautiful, outrageous Émilie du Châtelet and the charmingly subversive Mary Somerville. Against great odds, Émilie and Mary taught themselves mathematics, and did it so well that they each became a world authority on Newtonian mathematical physics.Seduced by Logic begins with Émilie du Châtelet, an 18th-century French aristocrat, intellectual, and Voltaire''s lover, whose true ambition was to be a mathematician. She strove not only to further Newton''s ideas in France, but to prove that they had French connections, including to the work of Descartes, whom Newton had read. She translated the great Principia Mathematica into French, in what became the accepted French version of Newton''s work, and was instrumental in bringing Newton''s revolutionary opus to a Continental audience. A century later, in Scotland, Mary STrade Review...timely reminder of how little things have changed since the 19th century and how much women of science can accomplish. * Wall Street Journal *Table of ContentsIntroduction ; 1 Madame Newton du Chatelet ; 2 Creating the theory of gravity: the Newtonian controversy ; 3 Learning mathematics and fighting for freedom ; 4 Emilie and Voltaire's Academy of Free Thought ; 5 Testing Newton: the'New Argonauts' ; 6 The danger in Newton: life, love and politics ; 7 The nature of light: Emilie takes on Newton ; 8 Searching for 'energy': Emilie discovers Leibniz ; 9 Mathematics and free will ; 10 The re-emergence of Madame Newton du Chatelet ; 11 Love letters to Saint-Lambert ; 12 Mourning Emilie ; 13 Mary Fairfax Somerville ; 14 The long road to fame ; 15 Mechanism of the Heavens ; 16 Mary's second book: popular science in the nineteenth century ; 17 Finding light waves: the 'Newtonian Revolution' comes of age ; 18 Mary Somerville: a fortunate life ; Epilogue: Declaring a point of view

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Book SynopsisGraph Theory.- Combinatorics.- Infinite Combinatorics and Graphs.Trade ReviewFrom the reviews:SIAM REVIEW"The narrative and proofs are well written, and the authors are given to frequent uses of humor. Students should find this book as easy to read as any other good-quality text written with them in mind. Each of the three chapters concludes with several paragraphs describing an excellent selection of more advanced texts or papers to consider for further study"From the reviews of the second edition: “Any undergraduate work in combinatorics or graph theory, whether a course or independent study, would likely be well served by this textbook … . The authors offer a wide selection of topics, often in more depth than other undergraduate texts, in an engaging and clear style. … Each chapter concludes with extensive notes on further reading.” (Brian Hopkins, Mathematical Reviews, Issue 2010 b) “Combinatorics and Graph Theory is a popular pair of topics to choose for an undergraduate course. … The book is written in a reader-friendly style and there are enough exercises. … It is certainly good that someone took the effort to write … in a form that is appropriate for undergraduates. … the book will most often be used for a reading class by a student who already has a background in combinatorics and who wants to learn about the set theoretical aspect of it.” (Miklós Bóna, SIGACT News, Vol. 40 (3), 2009)“This undergraduate textbook contains three chapters: Graph Theory, Combinatorics and Infinite Combinatorics and Graphs. … There is a short section on References in each chapter introducing briefly other books dealing with the topics covered in the respective chapter. A full list of 293 references, about 550 exercises and an index with 13 pages are also provided.” (Dalibor Froncek, Zentralblatt MATH, Vol. 1170, 2009)Table of ContentsGraph Theory.- Combinatorics.- Infinite Combinatorics and Graphs.

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Book Synopsis1 The Axiom of Extension.- 2 The Axiom of Specification.- 3 Unordered Pairs.- 4 Unions and Intersections.- 5 Complements and Powers.- 6 Ordered Pairs.- 7 Relations.- 8 Functions.- 9 Families.- 10 Inverses and Composites.- 11 Numbers.- 12 The Peano Axioms.- 13 Arithmetic.- 14 Order.- 15 The Axiom of Choice.- 16 Zorn's Lemma.- 17 Well Ordering.- 18 Transfinite Recursion.- 19 Ordinal Numbers.- 20 Sets of Ordinal Numbers.- 21 Ordinal Arithmetic.- 22 The Schröder-Bernstein Theorem.- 23 Countable Sets.- 24 Cardinal Arithmetic.- 25 Cardinal Numbers.Trade ReviewFrom the reviews:“This book is a very specialized but broadly useful introduction to set theory. It is aimed at ‘the beginning student of advanced mathematics’ … who wants to understand the set-theoretic underpinnings of the mathematics he already knows or will learn soon. It is also useful to the professional mathematician who knew these underpinnings at one time but has now forgotten exactly how they go. … A good reference for how set theory is used in other parts of mathematics … .” (Allen Stenger, The Mathematical Association of America, September, 2011)Table of Contents1 The Axiom of Extension.- 2 The Axiom of Specification.- 3 Unordered Pairs.- 4 Unions and Intersections.- 5 Complements and Powers.- 6 Ordered Pairs.- 7 Relations.- 8 Functions.- 9 Families.- 10 Inverses and Composites.- 11 Numbers.- 12 The Peano Axioms.- 13 Arithmetic.- 14 Order.- 15 The Axiom of Choice.- 16 Zorn’s Lemma.- 17 Well Ordering.- 18 Transfinite Recursion.- 19 Ordinal Numbers.- 20 Sets of Ordinal Numbers.- 21 Ordinal Arithmetic.- 22 The Schröder-Bernstein Theorem.- 23 Countable Sets.- 24 Cardinal Arithmetic.- 25 Cardinal Numbers.

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Book Synopsis1 Naive Set Theory.- 1.1 What is a Set?.- 1.2 Operations on Sets.- 1.3 Notation for Sets.- 1.4 Sets of Sets.- 1.5 Relations.- 1.6 Functions.- 1.7 Well-Or der ings and Ordinals.- 1.8 Problems.- 2 The ZermeloFraenkel Axioms.- 2.1 The Language of Set Theory.- 2.2 The Cumulative Hierarchy of Sets.- 2.3 The ZermeloFraenkel Axioms.- 2.4 Classes.- 2.5 Set Theory as an Axiomatic Theory.- 2.6 The Recursion Principle.- 2.7 The Axiom of Choice.- 2.8 Problems.- 3 Ordinal and Cardinal Numbers.- 3.1 Ordinal Numbers.- 3.2 Addition of Ordinals.- 3.3 Multiplication of Ordinals.- 3.4 Sequences of Ordinals.- 3.5 Ordinal Exponentiation.- 3.6 Cardinality, Cardinal Numbers.- 3.7 Arithmetic of Cardinal Numbers.- 3.8 Regular and Singular Cardinals.- 3.9 Cardinal Exponentiation.- 3.10 Inaccessible Cardinals.- 3.11 Problems.- 4 Topics in Pure Set Theory.- 4.1 The Borel Hierarchy.- 4.2 Closed Unbounded Sets.- 4.3 Stationary Sets and Regressive Functions.- 4.4 Trees.- 4.5 Extensions of Lebesgue Measure.- 4.6 A ReTable of ContentsPreface; 1. Naïve Set Theory; 2. The Zermelo-Fraenkel Axioms; 3. Ordinal and Cardinal Numbers; 4. Topics in Pure Set Theory; 5. The Axiom of Constructibility; 6. Independence Proofs in Set Theory; 7. Non-Well-Founded Set Theory; Bibliography; Glossary of Symbols; Index

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Book SynopsisThe later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.Table of Contents1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem “P ? NP?”.- 8 Newton’s Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bézout’s Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

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Book SynopsisAttempts to give a systematic presentation of both logic and type theory from a categorical perspective, using the unifying concept of fibred category. This book is useful for logicians, type theorists, category theorists and (theoretical) computer scientists.Trade Review"The author's achievement in collecting and organizing a very large body of material in coherent form,... this is first and foremost an encyclopaedic work, into which specialists will delve with much pleasure and profit... One very welcome feature of the book is a comprehensive bibliography of nearly 350 items..." --Zentralblatt für Mathematik, vol.905R.A.G. Seely"This book will be the standard reference in its field for some time to come." --The Bulletin of Symbolic Logic, Vol. 6Table of ContentsChapter Headings only. Preface. Contents. Preliminaries. Prospectus. Introduction to fibred category theory. Simple type theory. Equational logic. First order predicate logic. Higher order predicate logic. The effective topos. Internal category theory. Polymorphic type theory. Advanced fibred category theory. First order dependent type theory. Higher order dependent type theory. References. Notation index. Subject index.

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Book SynopsisClassical logic has proved inadequate in various areas of computer science, artificial intelligence, mathematics, philosopy and linguistics. This is an introduction to extensions of first-order logic, based on the principle that many-sorted logic (MSL) provides a unifying framework in which to work.Table of Contents1. Standard second order logic; 2. Deductive calculi; 3. Categoricity of second order peano arithmetic; 4. Frames and general structures; 5. Type theory; 6. Many-sorted logic; 7. Applying MSL.

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Book SynopsisPetri nets are a popular and powerful formal model for the analysis and modelling of concurrent systems. The text is very clearly organised, with every notion carefully explained and every result proved. Clear exposition is given for place invariants, siphons, traps and many other important analysis techniques.Table of Contents1. Introduction; 2. Analysis techniques for petri nets; 3. S-systems and T-systems; 4. Liveness in free choice; 5. The coverability theorems; 6. The rank theorem; 7. Reduction and synthesis; 8. Home markings; 9. Reachability and shortest sequences; 10. Generalizations; Index; List of symbols; List of main results.

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Book SynopsisType theory is one of the most important tools in the design of higher-level programming languages, such as ML. This book introduces and teaches its techniques by focusing on one particularly neat system and studying it in detail. In this way, all the key ideas are covered without getting involved in the complications of more advanced systems, but concentrating rather on the principles that make the theory work in practice. This book takes a type-assignment approach to type theory, and the system considered is the simplest polymorphic one. The author covers all the basic ideas, including the system's relation to propositional logic, and gives a careful treatment of the type-checking algorithm which lies at the heart of every such system. Also featured are two other interesting algorithms that have been buried in inaccessible technical literature. The mathematical presentation is rigorous but clear, making the book at a level which can be used as an introduction to type theory for compTrade Review"This is an excellent introduction to type theory. It doesn't bog the reader down in any of the messy details of the proofs and yet it provides many of the most interesting results in the field....Overall, it is a great book for someone who wants to get his feet wet in type theory, but doesn't want to get in over his head." Sigact News"...the book makes useful and stimulating reading and it will be an essential tool for computer scientists working in type theory and related areas." Mathematical ReviewsThe proofs in this book are given in great detail, and still the author succeeds in writing the book in a clear but not too technical style. It is easy and pleasurable to read this book." Journal of Symbolic LogicTable of ContentsIntroduction; 1. The type-free λ-calculus; 2. Assigning types to terms; 3. The principal-type algorithm; 4. Type assignment with equality; 5. A version using typed terms; 6. The correspondence with implication; 7. The converse principal-type algorithm; 8. Counting a type's inhabitants; 9. Technical details; Answers to starred exercises; Bibliography; Table of principal types; Index.

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Book SynopsisThis is a first course in propositional modal logic, suitable for mathematicians, computer scientists and philosophers. Emphasis is placed on semantic aspects, in the form of labelled transition structures, rather than on proof theory.Trade Review"This text should appeal to anyone with an interest in model logic...an attractive choice for self-study." J.M. Plotkin, Mathematical Reviews"...offers a distinctive viewpoint and is easy to learn from." D.V. Feldman, ChoiceTable of ContentsIntroduction; Acknowledgements; Part I. Preliminaries: 1. Survey of propositional logic; 2. The modal language; Part II. Transition Structures and Semantics: 3. Labelled transition structures; 4. Valuation and satisfaction; 5. Correspondence theory; 6. The general confluence result; Part III. Proof Theory and Completeness: 7. Some consequence relations; 8. Standard formal systems; 9. The general completeness result; 10. Kripke-completeness; Part IV. Model Constructions: 11. Bismulations; 12. Filtrations; 13. The finite model property; Part V. More Advanced Material: 14. SLL logic; 15. Löb logic; 16. Canonicity without the fmp; 17. Transition structures aren't enough; Part VI. Two Appendices: Bibliography.

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Book SynopsisThis book describes the mathematical aspects of the semantics of programming languages. The main goals are to provide formal tools to assess the meaning of programming constructs in ways independent of both language and platform. To do this the authors explain denotational and operational semantics and exploit the duality between them.Trade ReviewReview of the hardback: '… extremely useful as a reference for classical results and techniques in general domain theory, very balanced in the treatment of mathematical properties of domains and their operational motivations and applications, and always deep in the choice of topics and in unravelling the connections between them.' Felice Cardone, Science of Computer ProgrammingReview of the hardback: 'This is an excellent, thorough monograph … a rich and comprehensive source of information, it is very useful as a reference to classical results in domain theory and lambda calculus.' Paula G. Severi, Zentralblatt MATHTable of ContentsPreface; Notation; 1. Continuity and computability; 2. Syntactic theory of λ-calculus; 3. D∞ models and intersection types; 4. Interpretation of λ-calculi in CCC's; 5. CCC's of algebraic dcpo's; 6. The language PCF; 7. Domain equations; 8. Values and computations; 9. Powerdomains; 10. Stone duality; 11. Dependent and second order types; 12. Stability; 13. Towards linear logic; 14. Sequentiality; 15. Domains and realizability; 16. Functions and processes; Appendix 1: summary of recursion theory; Appendix 2: summary of category theory; References and bibliography; Index.

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Book SynopsisIn this 1987 text Professor Jech gives a unified treatment of the various forcing methods used in set theory, and presents their important applications. Product forcing, iterated forcing and proper forcing have proved powerful tools when studying the foundations of mathematics, for instance in consistency proofs. The book is based on graduate courses though some results are also included, making the book attractive to set theorists and logicians.Table of ContentsPreface; Part I. Product Forcing: 1. Forcing and Boolean-valued models; 2. Properties of the generic extension; 3. Examples of generic reals; 4. Product forcing; 5. Examples of product forcing; 6. The Lévy collapse; 7. Product measure forcing; Part II. Iterated Forcing: 8. Two step iteration; 9. Finite support iteration; 10. Martin's axiom; 11. Suslin's problem; 12. Whitehead's problem; 13. Kaplansky's conjecture; 14. Countable support iteration; 15. Borel's conjecture; Part III. Proper Forcing: 16. Stationary sets; 17. Infinite games on complete Boolean algebras; 18. Proper forcing; 19. Examples of proper forcing; 20. Iteration of proper forcing; 21. The Proper Forcing Axiom; 22. Martin's maximum; 23. Well-founded iteration; Bibliography; Index of symbols; Subject index; Author index.

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Book SynopsisDomain theory is an established part of theoretical computer science, used in giving semantics to programming languages and logics. In mathematics and logic it has also proved to be useful in the study of algorithms. This book is devoted to providing a unified and self-contained treatment of the subject. The theory is presented in a mathematically precise manner which nevertheless is accessible to mathematicians and computer scientists alike. The authors begin with the basic theory including domain equations, various domain representations and universal domains. They then proceed to more specialized topics such as effective and power domains, models of lambda-calculus and so on. In particular, the connections with ultrametric spaces and the KleeneâKreisel continuous functionals are made precise. Consequently the text will be useful as an introductory textbook (earlier versions have been class-tested in Uppsala, Gothenburg, Passau, Munich and Swansea), or as a general reference for profTrade ReviewReview of the hardback: '… can be thoroughly recommended for anyone interested in computability.' Steven Vickers, The Computer JournalTable of ContentsPreliminaries; Part I. Basic Theory: 1. Fixed points; 2. Complete partial orders; 3. Domains; 4. Domain equations; 5. Topology; 6. Representation theory; 7. A universal domain; Part II. Special Topics: 8. Representability in domains; 9. Basic recursion theory; 10. Effective domains; 11. Power domains; 12. Domains as models of formal theories; References; Index of symbols; Index.

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Book SynopsisThis is an up-to-date and integrated introduction to model theory, designed to be used for graduate courses (for students who are familiar with first-order logic), and as a reference for more experienced logicians and mathematicians. Model theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide variety of other areas such as set theory, geometry, algebra (in particular group theory), and computer science (e.g. logic programming and specification). Professor Hodges emphasises definability and methods of construction, and introduces the reader to advanced topics such as stability. He also provides the reader with much historical information and a full bibliography, enhancing the book's use as a reference.Trade Review"...sets forth the basic tools that the fledgling model theorist should master....a new standard reference in this subject." J.M. Plotkin, Mathematical Reviews"...an important contribution to the literature...one of the few (or perhaps the only) books in print that develop the essentials of model theory in the modern style...valuable for its novel treatment and exposition of some important topics...Throughout the book, Hodges's vast knowledge of logic impresses itself on the reader. There is a beautiful variety and mix of classical and more recent techniques and ideas, especially in applications. It is a pleasure to leaf through the book..." Anand Pillay, Jrl of Symbolic LogicTable of ContentsPart I. Language and Structure: 1. Naming of parts; 2. Classifying structures; 3. Structures that look alike; 4. Automorphisms; 5. Interpretations; Part II. Piecing Together a Structure: 6. The first order case; 7. The countable case; 8. The existential case; 9. The Horn case; Part III. Consequences of Compactness: 10. Saturation; 11. Combinatorial; 12. Expansions and categoricity; Examples; Bibliography; Index.

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Book SynopsisThe Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency.Table of Contents1. G and other normal modal propositional logics; 2. Peano Arithmetic; 3. The box as Bew; 4. Some applications of G; 5. Semantics for G and other modal logics; 6. Canonical models; 7. The completeness and decidability of G; 8. Trees for G; 9. Calculating the truth-values of fixed points; 10. Rosser's theorem; 11. The fixed-point theorem; 12. Solovay's completeness theorems; 13. An S4-preserving proof-theoretical treatment of modality; 14. The Craig Interpolation Lemma for G.

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Book SynopsisMultiple-conclusion logic extends formal logic by allowing arguments to have a set of conclusions instead of a single one, the truth lying somewhere among the conclusions if all the premises are true. The extension opens up interesting possibilities based on the symmetry between premises and conclusions, and can also be used to throw fresh light on the conventional logic and its limitations.Table of ContentsPreface; Introduction; Part I. Multiple and Single Conclusions; 1. Single-conclusion calculi; 2. Multiple-conclusion calculi; 3. Tree proofs; 4. Axiomatisability; 5. Counterparts; 6. Infinite rules; Part II. Graph Proofs; 7. Graph arguments; 8. Kneale proofs; 9. Cross-reference; 10. Abstract proofs; 11. Single-conclusions proofs; 12. Infinite proofs; Part III: Many-valued Logic; 13. Many-valued calculi; 14. matrices; 15. Many-valuedness; 16. Counterparts; 17. Categoricity; 18. Two-valued logic; 19. Axiomatisation; Part IV. Natural Deduction; 20. Natural Deduction; Bibliography; Index.

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Book SynopsisDiagrams are widely used in reasoning about problems in physics, mathematics and logic, but have traditionally been considered to be only heuristic tools and not valid elements of mathematical proofs. This book challenges this prejudice against visualisation in the history of logic and mathematics and provides a formal foundation for work on natural reasoning in a visual mode.Trade Review"I will conclude by offering general praise for a work that really does deserve praise. Even more, it deserves to be read by those mathematicians and logicians who adhere to the general prejudice against diagrams. Shin has gone much farther than anyone in showing how a diagrammatic system can hold its own as a medium for reasoning....this book is clear and convincing....Shin's mastery and manipulation of her technical tools is always thorough and lucid. All in all, this is a very impressive, valuable piece of work." Modern Logic"The book is refreshing in its clarity and modesty, and in its regular return to historial precedents. It will give welcome reassurance to all those logicians for whom pictures seem indisputably the simplest way for the mind to comprehend syllogistic reasoning." M. Schiff, Choice"...The author's style is nice, very precise and approachable. The soundness and completeness of Venn systems provide a legitimate reason why logicians should be interested in nonlinguistic representation systems. Undoubtedly, this will be one of the rare inspiring sources motivating others to work in the field of visual representation systems." Branislav Boricic, Mathematical Reviews"Shin has shown that sysytems of diagrams can be developed that precisley parallel axiomatic formal systems for monadic predicate logic....Shin correctly points out what diagram systems and verbal systems can both do...that realistic pictures...can't." Phillip L. Peterson, Philosophy in ReviewTable of ContentsAcknowledgements; 1. Introduction; 2. Preliminaries; 3. Venn-I; 4. Venn-II; 5. Venn-II and L0; 6. Diagrammatic versus linguistic representation; 7. Conclusion; Appendix; References; Index.

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Book SynopsisKurt GÃdel (1906â1978) did groundbreaking work that transformed logic and other important aspects of our understanding of mathematics, especially his proof of the incompleteness of formalized arithmetic. This book on different aspects of his work and on subjects in which his ideas have contemporary resonance includes papers from a May 2006 symposium celebrating GÃdel's centennial as well as papers from a 2004 symposium. Proof theory, set theory, philosophy of mathematics, and the editing of GÃdel's writings are among the topics covered. Several chapters discuss his intellectual development and his relation to predecessors and contemporaries such as Hilbert, Carnap, and Herbrand. Others consider his views on justification in set theory in light of more recent work and contemporary echoes of his incompleteness theorems and the concept of constructible sets.Table of ContentsPart I. General: 1. The Gödel editorial project: a synopsis Solomon Feferman; 2. Future tasks for Gödel scholars John W. Dawson, Jr, and Cheryl A. Dawson; Part II. Proof Theory: 3. Kurt Gödel and the metamathematical tradition Jeremy Avigad; 4. Only two letters: the correspondence between Herbrand and Gödel Wilfried Sieg; 5. Gödel's reformulation of Gentzen's first consistency proof for arithmetic: the no-counter-example interpretation W. W. Tait; 6. Gödel on intuition and on Hilbert's finitism W. W. Tait; 7. The Gödel hierarchy and reverse mathematics Stephen G. Simpson; 8. On the outside looking in: a caution about conservativeness John P. Burgess; Part III. Set Theory: 9. Gödel and set theory Akihiro Kanamori; 10. Generalizations of Gödel's universe of constructible sets Sy-David Friedman; 11. On the question of absolute undecidability Peter Koellner; Part IV. Philosophy of Mathematics: 12. What did Gödel believe and when did he believe it? Martin Davis; 13. On Gödel's way in: the influence of Rudolf Carnap Warren Goldfarb; 14. Gödel and Carnap Steve Awodey and A. W. Carus; 15. On the philosophical development of Kurt Gödel Mark van Atten and Juliette Kennedy; 16. Platonism and mathematical intuition in Kurt Gödel's thought Charles Parsons; 17. Gödel's conceptual realism Donald A. Martin.

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Book SynopsisThis book presents a unifying framework for using priority arguments to prove theorems in computability. Priority arguments provide the most powerful theorem-proving technique in the field, but most of the applications of this technique are ad hoc, masking the unifying principles used in the proofs. The proposed framework presented isolates many of these unifying combinatorial principles and uses them to give shorter and easier-to-follow proofs of computability-theoretic theorems. Standard theorems of priority levels 1, 2, and 3 are chosen to demonstrate the framework's use, with all proofs following the same pattern. The last section features a new example requiring priority at all finite levels. The book will serve as a resource and reference for researchers in logic and computability, helping them to prove theorems in a shorter and more transparent manner.Table of Contents1. Introduction; 2. Systems of trees of strategies; 3. Σ1 constructions; 4. Δ2 constructions; 5. µ2 constructions; 6. Δ3 constructions; 7. Σ3 constructions; 8. Paths and links; 9. Backtracking; 10. Higher level constructions; 11. Infinite systems of trees.

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Book SynopsisWhat are the appropriate axioms for mathematics? Through a series of case studies, this volume examines these axioms to prove particular theorems in core areas including algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics.Table of ContentsList of tables; Preface; Acknowledgements; 1. Introduction; Part I. Development of Mathematics within Subsystems of Z2: 2. Recursive comprehension; 3. Arithmetical comprehension; 4. Weak König's lemma; 5. Arithmetical transfinite recursion; 6. π11 comprehension; Part II. Models of Subsystems of Z2: 7. β-models; 8. ω-models; 9. Non-ω-models; Part III. Appendix: 10. Additional results; Bibliography; Index.

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Book SynopsisThis book gives an introduction to theories of computability from a mathematically sophisticated point of view. It treats not only 'the' theory of computability, but also a variety of other theories. Each topic is taken from the beginning and developed to a point at which the results can be appreciated.Trade Review"The author is extremely precise and meticulous, giving as much care to a proof that any context-free language can be generated by a grammar in Greibach normal form (GNF) as to a priority argument." Computing Reviews"...an excellent textbook for a beginning researcher who wants to familiarize himself/herself with several axiomatic frameworks for the theory of computability...I enjoyed the nuggets of wit spread throughout the book, providing a pleasant contrast to the abstract material." SIGACT News"Aiming readers at the frontier of research, the book has a point to make, and it makes it in an elegant and attractive way." Mathematical Reviews"The book, which has a list of about 200 references, is well indexed, well organized, and quite well written. But the strongest commendation is for its selection of material and its coherent organization." Siam ReviewTable of ContentsPreface; 1. Finite functions and relations; 2. Finite automata and their languages; 3. Grammars and their languages; 4. Computable functions and relations; References; Author index; Subject index; Math index.

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Book SynopsisThis book introduces a new approach to building models of bounded arithmetic, with techniques drawn from recent results in computational complexity. Propositional proof systems and bounded arithmetics are closely related. In particular, proving lower bounds on the lengths of proofs in propositional proof systems is equivalent to constructing certain extensions of models of bounded arithmetic. This offers a clean and coherent framework for thinking about lower bounds for proof lengths, and it has proved quite successful in the past. This book outlines a brand new method for constructing models of bounded arithmetic, thus for proving independence results and establishing lower bounds for proof lengths. The models are built from random variables defined on a sample space which is a non-standard finite set and sampled by functions of some restricted computational complexity. It will appeal to anyone interested in logical approaches to fundamental problems in complexity theory.Trade Review"Jan Krajíček is the leading expert on these problems and in this book he provides a new approach to builing models of bounded arithmetic which combines methods and techniques from model theory, forcing and computational complexity. Personally, I find Krajíček's approach a highly stimulating collage of ideas. I recommend this book strongly to anyone interested in logical approaches to fundamental problems in complexity theory." Soren M. Riis for Mathematical ReviewsTable of ContentsPreface; Acknowledgements; Introduction; Part I. Basics: 1. The definition of the models; 2. Measure on β; 3. Witnessing quantifiers; 4. The truth in N and the validity in K(F); Part II. Second Order Structures: 5. Structures K(F,G); Part III. AC0 World: 6. Theories IΔ0, IΔ0(R) and V10; 7. Shallow Boolean decision tree model; 8. Open comprehension and open induction; 9. Comprehension and induction via quantifier elimination: a general reduction; 10. Skolem functions, switching lemma, and the tree model; 11. Quantifier elimination in K(Ftree,Gtree); 12. Witnessing, independence and definability in V10; Part IV. AC0(2) World: 13. Theory Q2V10; 14. Algebraic model; 15. Quantifier elimination and the interpretation of Q2; 16. Witnessing and independence in Q2V10; Part V. Towards Proof Complexity: 17. Propositional proof systems; 18. An approach to lengths-of-proofs lower bounds; 19. PHP principle; Part VI. Proof Complexity of Fd and Fd(+): 20. A shallow PHP model; 21. Model K(Fphp,Gphp) of V10; 22. Algebraic PHP model?; Part VII. Polynomial-Time and Higher Worlds: 23. Relevant theories; 24. Witnessing and conditional independence results; 25. Pseudorandom sets and a Löwenheim–Skolem phenomenon; 26. Sampling with oracles; Part VIII. Proof Complexity of EF and Beyond: 27. Fundamental problems in proof complexity; 28. Theories for EF and stronger proof systems; 29. Proof complexity generators: definitions and facts; 30. Proof complexity generators: conjectures; 31. The local witness model; Appendix. Non-standard models and the ultrapower construction; Standard notation, conventions and list of symbols; References; Name index; Subject index.

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Book SynopsisThis two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume 1 includes formal proof techniques, a section on applications of compactness (including nonstandard analysis), a generous dose of computability and its relation to the incompleteness phenomenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen theorem.Trade ReviewReview of the hardback: ' … stands among the dozen or so books that will remain within easy reach from my desk … a highly worthwhile addition to the literature.' The Review of Modern LogicTable of ContentsPreface; 1. Basic logic; 2. The second incompleteness theorem; Bibliography; List of symbols; Index.

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Book SynopsisThis two-volume work bridges the gap between introductory expositions of logic or set theory on one hand, and the research literature on the other. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly conversational lecture style that makes them equally effective for self-study or class use. Volume II, on formal (ZFC) set theory, incorporates a self-contained 'chapter 0' on proof techniques so that it is based on formal logic, in the style of Bourbaki. The emphasis on basic techniques will provide the reader with a solid foundation in set theory and provides a context for the presentation of advanced topics such as absoluteness, relative consistency results, two expositions of Godel's constructible universe, numerous ways of viewing recursion, and a chapter on Cohen forcing.Trade Review"many readers, especially those with a philosophical bent, will find this book a good choice for the classroom." - Richard E. Hodel, Duke UniversityTable of ContentsPreface; 1. A bit of logic: a user's toolbox; 2. The set-theoretic universe, naively; 3. The axioms of set theory; 4. The axiom of choice; 5. The natural numbers; transitive closure; 6. Order; 7. Cardinality; 8. Forcing; Bibliography; List of symbols; Index.

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Book SynopsisThis 2001 book presents a unified approach to the foundations of mathematics in the theory of sets, covering conventional and finitary mathematics. It analyses the relation between the concepts of 'natural number' and 'set', and investigates the logic of quantification over the universe of sets. Suitable for both philosophers and mathematicians.Trade ReviewReview of the hardback: ' … this book is thought-provoking … distinctive approach to the twin issues of mathematical ontology and mathematical foundations'. Australasian Journal of PhilosophyTable of ContentsPreface; Part I. Preliminaries: 1. The idea of foundations of mathematics; 2. Simple arithmetic; Part II. Basic Set Theory: 3. Semantics, ontology and logic; 4. The principal axioms and definitions of set theory; Part III. Cantorian Set Theory: 5. Cantorian finitism; 6. The axiomatic method; 7. Axiomatic set theory; Part IV. Euclidean Set Theory: 8. Euclidian finitism; 9. The Euclidean theory of cardinality; 10. The theory of simply infinite systems; 11. Euclidean set theory from the Cantorian standpoint; 12. Envoi; Appendices; Bibliography; Index.

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Book SynopsisIn this 1987 text Professor Jech gives a unified treatment of the various forcing methods used in set theory, and presents their important applications. Product forcing, iterated forcing and proper forcing have proved powerful tools when studying the foundations of mathematics, for instance in consistency proofs. The book is based on graduate courses though some results are also included, making the book attractive to set theorists and logicians.Table of ContentsPreface; Part I. Product Forcing: 1. Forcing and Boolean-valued models; 2. Properties of the generic extension; 3. Examples of generic reals; 4. Product forcing; 5. Examples of product forcing; 6. The Lévy collapse; 7. Product measure forcing; Part II. Iterated Forcing: 8. Two step iteration; 9. Finite support iteration; 10. Martin's axiom; 11. Suslin's problem; 12. Whitehead's problem; 13. Kaplansky's conjecture; 14. Countable support iteration; 15. Borel's conjecture; Part III. Proper Forcing: 16. Stationary sets; 17. Infinite games on complete Boolean algebras; 18. Proper forcing; 19. Examples of proper forcing; 20. Iteration of proper forcing; 21. The Proper Forcing Axiom; 22. Martin's maximum; 23. Well-founded iteration; Bibliography; Index of symbols; Subject index; Author index.

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Book SynopsisA crystal clear introduction to category theory that demystifies functors, natural transformations, limits and colimits, adjunctions and more. Any beginning postgraduate mathematician will find all they need in this excellent text to access the subject. Over 200 exercises are provided with solutions available online.Trade Review"This textbook presents a useful introduction to basic category theory, and would be suitable for a first course at the undergraduate level in computer science or mathematics." Steve Awodey, Mathematical ReviewsTable of ContentsPreface; 1. Categories; 2. Basic gadgetry; 3. Functors and natural transformations; 4. Limits and colimits in general; 5. Adjunctions; 6. Posets and monoid sets; Bibliography; Index.

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Book SynopsisModel theory is concerned with the notions of definition, interpretation and structure in a very general setting, and is applied to a wide range of other areas such as set theory, geometry, algebra and computer science. This book provides an integrated introduction to model theory for graduate students.Trade Review"...sets forth the basic tools that the fledgling model theorist should master....a new standard reference in this subject." J.M. Plotkin, Mathematical Reviews"...an important contribution to the literature...one of the few (or perhaps the only) books in print that develop the essentials of model theory in the modern style...valuable for its novel treatment and exposition of some important topics...Throughout the book, Hodges's vast knowledge of logic impresses itself on the reader. There is a beautiful variety and mix of classical and more recent techniques and ideas, especially in applications. It is a pleasure to leaf through the book..." Anand Pillay, Jrl of Symbolic LogicTable of ContentsPart I. Language and Structure: 1. Naming of parts; 2. Classifying structures; 3. Structures that look alike; 4. Automorphisms; 5. Interpretations; Part II. Piecing Together a Structure: 6. The first order case; 7. The countable case; 8. The existential case; 9. The Horn case; Part III. Consequences of Compactness: 10. Saturation; 11. Combinatorial; 12. Expansions and categoricity; Examples; Bibliography; Index.

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Book SynopsisThe interplay between model theory and other branches of mathematics has led to intriguing results. In this book, the theme is the interplay between model theory and the theory of modules. It is an introduction to the subject and introduces the requisite model theory and module theory as is needed.Trade Review"...absolutely essential to anyone with any interest in the subject matter, because of its comprehensiveness, currency, well-written style, and extensive bibliographical references." Journal of Symbolic Logic"...contains a rich collection of concrete examples and exercises. It may be useful both as a reference book and as a texbook." Mathematical ReviewsTable of ContentsIntroduction; Acknowledgements; Notations and conventions; Remarks on the development of the area; Section summaries; 1. Some preliminaries; 2. Positive primitive formulas and the sets they define; 3. Stability and totally transcendental modules; 4. Hulls; 5. Forking and ranks; 6. Stability-theoretic properties of types; 7. Superstable modules; 8. The lattice of pp-types and free realisations of pp-types; 9. Types and the structure of pure-injective modules; 10. Dimension and decomposition; 11. Modules over artinian rings; 12. Functor categories; 13. Modules over artin algebras; 14. Projective and flat modules; 15. Torsion and torsionfree classes; 16. Elimination of quantifiers; 17. Decidability and undecidability; Problems page; Bibliography; Examples index; Notation index; Index.

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Book SynopsisThis is an introductory textbook which is designed to be useful not only to intending logicians but also to mathematicians in general.Table of ContentsPreface; 1. Informal statement calculus; 2. Formal statement calculus; 3. Informal predicate calculus; 4. Formal predicate calculus; 5. Mathematical systems; 6. The Gödel incompleteness theorem; 7. Computability, unsolvability, undecidability; Appendix; Hints and solutions to selected exercises; References and further reading; Glossary of symbols; Index.

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Book SynopsisThe study of permutations groups has always been closely associated with that of highly symmetric structures. The objects considered here are countably infinite, but have only finitely many different substructures of any given finite size. This book discusses such structures, their substructures and their automorphism groups using a wide range of techniques.Trade Review' … a very attractive textbook for any student interested in discrete mathematics.' International Mathematical NewsTable of Contents1. Introduction; 2. Preliminaries; 3. Examples and growth rates; 4. Subgroups; 5. Miscellaneous topics.

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Book SynopsisA study of techniques for formal theorem-proving, with particular reference to Cambridge LCF (Logic for Computable Functions). The book is aimed at graduate students and researchers in theoretical computer science.Trade Review"This book is well-written: it is a good text for any reader who wants to become familiar with Cambridge LCF, or, in general, with machine assisted (formal) proof construction." Mathematical ReviewsTable of ContentsPart I. Preliminaries: 1. Survey and history of LCF; 2. Formal proof in first order logic; 3. A logic of computable functions; 4. Structural induction; Part II. Cambridge LCF: 5. Syntactic operators for PPL; 6. Theory structure; 7. Axioms and interference rules; 8. Tactics and tacticals; 9. Rewriting and simplification; 10. Sample proofs; Bibliography; Index.

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Book SynopsisThe theory of sets of multiples, a subject which lies at the intersection of analytic and probabilistic number theory, has seen much development since the publication of Sequences by Halberstam and Roth nearly thirty years ago. The area is rich in problems, many of them still unsolved or arising from current work. The author sets out to give a coherent, essentially self-contained account of the existing theory and at the same time to bring the reader to the frontiers of research. One of the fascinations of the theory is the variety of methods applicable to it, which include Fourier analysis, group theory, high and ultra-low moments, probability and elementary inequalities, as well as several branches of number theory. This Tract is the first devoted to the subject, and will be of value to research workers or graduate students in number theory.Trade Review"This book...should make a useful addition to the libraries of those interested in this general subject area." G. Greaves, Mathematical ReviewsTable of ContentsPreface; Introduction; Notation; First ideas; 1. Besicovitch and Behrend sequences; 2. Derived sequences and densities; 3. Oscillation; 4. Probabilistic group theory; 5. Divisor density; 6. Divisor uniform distribution; 7. H(x,y,z); Bibliography; Index.

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Book SynopsisBy considering the size of the logical network needed to perform a given computational task, the intrinsic difficulty of that task can be examined. Boolean function complexity, the combinatorial study of such networks, is a subject that started back in the 1950s and has today become one of the most challenging and vigorous areas of theoretical computer science. The papers in this book stem from the London Mathematical Society Symposium on Boolean Function Complexity held at Durham University in July 1990. The range of topics covered will be of interest to the newcomer to the field as well as the expert, and overall the papers are representative of the research presented at the Symposium. Anyone with an interest in Boolean Function complexity will find that this book is a necessary purchase.Table of Contents1. Relationships between monotone and non-monotone network complexity; 2. On read-once Boolean functions; 3. Boolean function complexity: a lattice-theoretic perspective; 4. Monotone complexity; 5. On submodular complexity measures; 6. Why is Boolean complexity so difficult?; 7. The multiplicative complexity of Boolean quadratic forms; 8. Some problems involving Razborov–Smolensky polynomials; 9. Symmetry functions in AC0; 10. Boolean complexity and probabilistic constructions; 11. Networks computing Boolean functions for multiple input values; 12. Optimal carry save networks.

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Book SynopsisA collection of expository and research articles derived from the SERC 'Logic for IT' Summer School and Conference on Proof Theory.Table of ContentsPreface; Programme of lectures; 1. Basic proof theory S. Wainer and L. Wallen; 2. A short course in ordinal analysis W. Pohlers; 3. Proofs as programs H. Schwichtenberg; 4. A simplified version of local predicativity W. Buchholz; 5. A note on bootstrapping intuitionistic bounded arithmetic S. Buss; 6. Termination orderings and complexity characterisations E. Cichon; 7. Logics for termination and correctness of functional programs, II. Logics of strength PRA S. Feferman; 8. Reflecting the semantics of reflected proof D. Howe; 9. Fragments of Kripke-Platek set theory with infinity M. Rathjen; 10. Provable computable selection functions on abstract structures J. Tucker and J. Zucker.

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