Mathematical logic Books

Book SynopsisSurveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in each. This work considers every proposed fix, each with its distinctive philosophical advantages and drawbacks.Trade ReviewCo-Winner of the 2007 Shoenfield Prize, Association for Symbolic Logic "Fixing Frege fills a serious gap in the Frege's literature (always increasing but perhaps with an excessive attention paid to semantics and the philosophy of language) and should remain for a long time a necessary reference for scholars in the field."--Ignacio Angelelli, Review of Modern LogicTable of ContentsAcknowledgments ix CHAPTER 1: Frege, Russell, and After 1 CHAPTER 2: Predicative Theories 86 CHAPTER 3: Impredicative Theories 146 Tables 215 Notes 227 References 241 Index 249

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Book SynopsisWhat did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? This book explains the history behind the development of our mathematical notation system.Trade Review"Mazur (Euclid in the Rainforest) gives readers the fascinating history behind the mathematical symbols we use, and completely take for granted, every day. Mathematical notation turns numbers into sentences--or, to the uninitiated, a mysterious and impenetrable code. Mazur says the story of math symbols begins some 3,700 years ago, in ancient Babylon, where merchants incised tallies of goods on cuneiform tablets, along with the first place holder--a blank space. Many early cultures used letters for both numbers and an alphabet, but convenient objects like rods, fingers, and abacus beads, also proved popular. Mazur shows how our 'modern' system began in India, picking up the numeral 'zero' on its way to Europe, where it came into common use in the 16th century, thanks to travelers and merchants as well as mathematicians like Fibonacci. Signs for addition, subtraction, roots, and equivalence followed, but only became standardized through the influence of scientists and mathematicians like Rene Descartes and Gottfried Leibniz. Mazur's lively and accessible writing makes what could otherwise be a dry, arcane history as entertaining as it is informative."--Publishers Weekly "[A] fascinating narrative... This is a nuanced, intelligently framed chronicle packed with nuggets--such as the fact that Hindus, not Arabs, introduced Arabic numerals. In a word: enlightening."--George Szpiro, Nature "Mazur begins by illustrating how the ancient Incas and Mayans managed to write specific, huge numbers. Then, for more than 200 pages, he traces the history of division signs, square roots, pi, exponents, graph axes and other symbols in the context of cognition, communication, and analysis."--Washington Post "Mazur delivers a solid exposition of an element of mathematics that is fundamental to its history."--Library Journal "Mazur treats only a subset of F. Cajori's monumental A History of Mathematical Notation (Dover, 1993 first edition 1922) and there is overlap with many other mathematical history books, but Mazur adds new findings and insights and it is so much more entertaining ... and these features make it an interesting addition to the existing literature for anybody with only a slight interest in mathematics or its history."--European Mathematical Society "Symbols like '+' and '=' are so ingrained that it's hard to conceive of math without them. But a new book, Enlightening Symbols: A Short History of Mathematical Notation and its Hidden Power, offers a surprising reminder: Until the early 16th century, math contained no symbols at all."--Kevin Hartnett, Boston Globe "Enlightening Symbols retraces the winding road that has led to the way we now teach, study, and conceive mathematics... Thanks to Mazur's playful approach to the subject, Enlightening Symbols offers an enjoyable read."--Gaia Donati, Science "If you enjoy reading about history, languages and science, then you'll enjoy this book... The best part is the writing is compelling enough that you don't have to be a mathematician to enjoy this informative book."--Guardian.com's GrrlScientist blog "[I]nformative, highly readable and scholarly."--Brian Rotman, Literary Review "[T]his insightful account of the historical development of a highly characteristic feature of the mathematical enterprise also represents a valuable contribution to our understanding of the nature of mathematics."--Eduard Glas, Mathematical Reviews Clippings "Joseph Mazur's beautiful book Enlightening Symbols tells the story of human civilization through the development of mathematical notation. Surprises abound... The book is visually exquisite, great care having been taken with illustrations and figures. Mazur's discussion of the emergence of particular symbols affords the reader an overview of the often difficult primary literature."--Donal O'Shea, Sarasota Herald-Tribune "At whatever depth one chooses to read it, Enlightening Symbols has something for everyone. It is entertaining and eclectic, and Mazur's personal and easy style helps connect us with those who led the long and winding search for the best ways to quantify and analyze our world. Their success has liberated us from 'the shackles of our physical impressions of space'--and of the particular and the concrete--'enabling imagination to wander far beyond the tangible world we live in, and into the marvels of generality.'"--Robyn Arianrhod, Notices of the Notices of the American Mathematical Society "Mazur introduces the reader to major characters, weaves in relevant aspects of wider culture and gives a feel for the breadth of mathematical history. It is a useful book for both student and interested layperson alike."--Mark McCartney, London Mathematical Society "[T]his is a good book. It is well written by an experienced author and is full of interesting facts about how the symbols used in mathematics have arisen. It would certainly interest anyone who studies the history of mathematics."--Phil Dyke, Leonardo "Mazur is a master story teller."--John Stillwell, Bulletin of the American Mathematical SocietyTable of ContentsIntroduction ix Definitions xxi Note on the Illustrations xxiii Part 1 Numerals 1 1. Curious Beginnings 3 2. Certain Ancient Number Systems 10 3. Silk and Royal Roads 26 4. The Indian Gift 35 5. Arrival in Europe 51 6. The Arab Gift 60 7. Liber Abbaci 64 8. Refuting Origins 73 Part 2 Algebra 81 9. Sans Symbols 85 10. Diophantus's Arithmetica 93 11. The Great Art 109 12. Symbol Infancy 116 13. The Timid Symbol 127 14. Hierarchies of Dignity 133 15. Vowels and Consonants 141 16. The Explosion 150 17. A Catalogue of Symbols 160 18. The Symbol Master 165 19. The Last of the Magicians 169 Part 3 The Power of Symbols 177 20. Rendezvous in the Mind 179 21. The Good Symbol 189 22. Invisible Gorillas 192 23. Mental Pictures 210 24. Conclusion 216 Appendix A Leibniz's Notation 221 Appendix B Newton's Fluxion of xn 223 Appendix C Experiment 224 Appendix D Visualizing Complex Numbers 228 Appendix E Quaternions 230 Acknowledgments 233 Notes 235 Index 269

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Book SynopsisIntroduces a theoretical entity: Agent_Zero. This title weaves a computational tapestry with threads from Plato, Hume, Darwin, Pavlov, Smith, Tolstoy, Marx, James, and Dostoevsky, among others.Trade Review"Agent Zero offers a solution to some of social science's great puzzles. Its behavioral basis is the interplay of emotion, cognition, and network contagion effects. It elegantly explains why so many human actions are so manifestly dysfunctional, and why some are downright evil."—George Akerlof, Nobel Laureate in Economics"Rarely has a book stimulated me intellectually as much as this one. Particularly exciting is the incorporation of agents who feel (affect) and deliberate, as well as influence one another through social interaction. Epstein is a brilliantly creative scholar and the range of applications showcased here is stunning. In sum, this is a pathbreaking book."—Paul Slovic, University of Oregon"Joshua Epstein proposes a parsimonious but powerful model of individual behavior that can generate an extraordinary range of group behaviors, including mob violence, manias and financial panics, rebellions, network dynamics, and a host of other complex social phenomena. This is a highly original, beautifully conceived, and important book."—Peyton Young, University of Oxford"In social science generally and most notably in economics, the rational actor model has long been the benchmark for policy analysis and institutional design. Epstein now offers a worthy alternative: Agent_Zero, a mathematically and computationally tractable agent whose inner workings are grounded in neuroscience. Much like you and me, Agent_Zero is influenced by emotion, reason, and social pressures. Epstein demonstrates that collections of Agent Zeros perform amazingly like real groups, teams, and societies and can therefore serve as the fundamental building blocks for what he calls Generative Social Science. The rational actor now has a true competitor. Agent_Zero is a major advance."—Scott Page, University of Michigan"This is social science based on how our brains actually work. Epstein's computerized 'agents' can feel passion and fear, and can influence each other emotionally. And when they interact, we see many of the realities of social life, from the dynamics of juries to racist violence to Arab springs. A remarkable and original piece of work."—W. Brian Arthur, Santa Fe InstituteTable of ContentsForeword xi Preface xiii Acknowledgments xv INTRODUCTION 1 MOTIVATION 1 Generate Social Dynamics 2 A Core Target 2 THE MODEL COMPONENTS 5 Model Overview 6 Skeletal Equation 8 Specific Components 9 ORGANIZATION 10 Part I: Mathematical Model 10 Part II: Agent-Based Model 11 Part III: Extensions 13 Replicability and Research Resources on the Princeton University Press Website 16 Part IV: Future Research and Conclusions 17 PART 1. MATHEMATICAL MODEL 19 I.1. THE PASSIONS: FEAR CONDITIONING 19 Fear Circuitry and the Perils of Fitness 20 Nomenclature of Conditioning 29 The Rescorla-Wagner Model 33 Social Examples 37 Fear Extinction 41 I.2. REASON: THE COGNITIVE COMPONENT 46 I.3. THE SOCIAL COMPONENT 51 Simple Version of the Core Target 55 Examples of Fear Contagion 57 Mechanisms of Fear Contagion 59 Conformist Empirical Estimates 63 Generalizing Rescorla-Wagner 67 The Central Case 69 Tolstoy: The First Agent Modeler 71 A Mathematical Aside on Social Norms as Vector Fields 74 Extinction of Majorities 78 I.4. INTERIM CONCLUSIONS 80 PART II. AGENT-BASED COMPUTATIONAL MODEL 81 Affective Component 84 "Rational" Component 85 Social Component 88 Action 89 Pseudocode 89 II.1. COMPUTATIONAL PARABLES 90 Parable 1: The Slaughter of Innocents through Dispositional Contagion 90 Parable 2: Agent_Zero Initiates: Leadership as Susceptibility to Dispositional Contagion 94 Run 3. Information Cuts Both Ways 96 Run 4. A Day in the Life of Agent_Zero: How Affect and Probability Can Change on Different Time Scales 98 Run 5. Lesion Studies 102 PART III. EXTENSIONS 107 III.1. ENDOGENOUS DESTRUCTIVE RADIUS 107 III.2. AGE AND IMPULSE CONTROL 109 III.3. FIGHT VS. FLIGHT 110 Case 1: Fight 111 Case 2: Flight 112 Capital Flight 114 III.4. REPLICATING THE Latane-DARLEY EXPERIMENT 114 Threshold Imputation 115 The Dialogue 118 III.5. MEMORY 118 III.6. COUPLINGS: ENTANGLEMENT OF PASSION AND REASON 122 Mathematical Treatment 124 III.7.ENDOGENOUS DYNAMICS OF CONNECTION STRENGTH 128 Affective Homophily 128 General Setup 130 Agent-Based Model: Nonequlibrium Dynamics 135 III.8. GROWING THE 2011 ARAB SPRING 138 III.9. JURY PROCESSES 143 Phase 1. Public Phase 143 Phase 2. Courtroom Trial Phase 145 Phase 3. Jury Phase 147 III.10. EMERGENT DYNAMICS OF NETWORK STRUCTURE 152 Network Structure Dynamics as a Poincare Map 153 Relation to Literature 159 III.11. MULTIPLE SOCIAL LEVELS 160 Agent_Zero as Witness to History 161 III.12. THE 18TH BRUMAIRE OF AGENT_ZERO 165 III.13. INTRODUCTION OF PRICES AND SEASONAL ECONOMIC CYCLES 168 Prices 168 A Christmas Story 173 III.14. SPIRALS OF MUTUAL ESCALATION 176 PART IV. FUTURE RESEARCH AND CONCLUSION 181 IV.1. FUTURE RESEARCH 181 IV.2. CONCLUSION 187 Civil Violence 187 Economics 188 Health Behavior 189 Psychology 190 Jury Dynamics 191 The Formation and Dynamics of Networks 191 Mutual Escalation Dynamics 192 Birth and Intergenerational Transmission 192 IV.3. TOWARD NEW GENERATIVE FOUNDATIONS 192 Appendix I. Threshold Imputation Bounds 195 Appendix II. Mathematica Code 197 Appendix III. Agent_Zero NetLogo Source Code 213 Appendix IV. Parameter Settings for Model Runs 221 References 227 Index 243

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Book SynopsisThis book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law'Trade Review"This is a marvelous and excellent introduction."--Adhemar Bultheel, European Mathematical Society Bulletin "A must-read for novices and experts alike. It can be used for a graduate-level topics course or as a reference text for researchers in the field. The exposition is outstanding, with hundreds of carefully chosen examples, figures and diagrams to illustrate the theory. For those who are up for a challenge, the book contains several open problems as well. An Introduction to Benford's Law will surely be the go-to text on the subject for years to come."--Pieter C. Allaart, Mathematical ReviewsTable of ContentsPreface vii 1 Introduction 1 1.1 History 3 1.2 Empirical evidence 4 1.3 Early explanations 6 1.4 Mathematical framework 7 2 Significant Digits and the Significand 11 2.1 Significant digits 11 2.2 The significand 12 2.3 The significand sigma-algebra 14 3 The Benford Property 22 3.1 Benford sequences 23 3.2 Benford functions 28 3.3 Benford distributions and random variables 29 4 The Uniform Distribution and Benford's Law 43 4.1 Uniform distribution characterization of Benford's law 43 4.2 Uniform distribution of sequences and functions 46 4.3 Uniform distribution of random variables 54 5 Scale-, Base-, and Sum-Invariance 63 5.1 The scale-invariance property 63 5.2 The base-invariance property 74 5.3 The sum-invariance property 80 6 Real-valued Deterministic Processes 90 6.1 Iteration of functions 90 6.2 Sequences with polynomial growth 93 6.3 Sequences with exponential growth 97 6.4 Sequences with super-exponential growth 101 6.5 An application to Newton's method 111 6.6 Time-varying systems 116 6.7 Chaotic systems: Two examples 124 6.8 Differential equations 127 7 Multi-dimensional Linear Processes 135 7.1 Linear processes, observables, and difference equations 135 7.2 Nonnegative matrices 139 7.3 General matrices 145 7.4 An application to Markov chains 162 7.5 Linear difference equations 165 7.6 Linear differential equations 170 8 Real-valued Random Processes 180 8.1 Convergence of random variables to Benford's law 180 8.2 Powers, products, and sums of random variables 182 8.3 Mixtures of distributions 202 8.4 Random maps 213 9 Finitely Additive Probability and Benford's Law 216 9.1 Finitely additive probabilities 217 9.2 Finitely additive Benford probabilities 219 10 Applications of Benford's Law 223 10.1 Fraud detection 224 10.2 Detection of natural phenomena 225 10.3 Diagnostics and design 226 10.4 Computations and Computer Science 228 10.5 Pedagogical tool 230 List of Symbols 231 Bibliography 234 Index 245

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Book SynopsisJohn Mather's seminal works in Hamiltonian dynamics represent some of the most important contributions to our understanding of the complex balance between stable and unstable motions in classical mechanics. His novel approach--known as Aubry-Mather theory--singles out the existence of special orbits and invariant measures of the system, which posseTable of ContentsPreface vii 1 Tonelli Lagrangians and Hamiltonians on Compact Manifolds 1 1.1 Lagrangian Point of View 1 1.2 Hamiltonian Point of View 4 2 From KAM Theory to Aubry-Mather Theory 8 2.1 Action-Minimizing Properties of Measures and Orbits on KAM Tori 8 3 Action-Minimizing Invariant Measures for Tonelli Lagrangians 18 3.1 Action-Minimizing Measures and Mather Sets 18 3.2 Mather Measures and Rotation Vectors 24 3.3 Mather's a-and B-Functions 28 3.4 The Symplectic Invariance of Mather Sets 35 3.5 An Example: The Simple Pendulum (Part I) 39 3.6 Holonomic Measures and Generic Properties of Tonelli Lagrangians 45 4 Action-Minimizing Curves for Tonelli Lagrangians 48 4.1 Global Action-Minimizing Curves: Aubry and Mane Sets 48 4.2 Some Topological and Symplectic Properties of the Aubry and Mane Sets 66 4.3 An Example: The Simple Pendulum (Part II) 68 4.4 Mather's Approach: Peierls' Barrier 71 5 The Hamilton-Jacobi Equation and Weak KAM Theory 76 5.1 Weak Solutions and Subsolutions of Hamilton-Jacobi and Fathi's Weak KAM theory 76 5.2 Regularity of Critical Subsolutions 85 5.3 Non-Wandering Points of the Mane Set 87 Appendices A On the Existence of Invariant Lagrangian Graphs 89 A.1 Symplectic Geometry of the Phase Space 89 A.2 Existence and Nonexistence of Invariant Lagrangian Graphs 91 B Schwartzman Asymptotic Cycle and Dynamics 97 B.1 Schwartzman Asymptotic Cycle 97 B.2 Dynamical Properties 99 Bibliography 107 Index 113

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Book SynopsisBetween inventing the concept of a universal computer in 1936 and breaking the German Enigma code during World War II, Alan Turing (1912-1954), the British founder of computer science and artificial intelligence, came to Princeton University to study mathematical logic. Some of the greatest logicians in the world--including Alonzo Church, Kurt GodeTrade Review"This book presents the story of Turing's work at Princeton University and includes a facsimile of his doctoral dissertation, 'Systems of Logic Based on Ordinals,' which he completed in 1936. The author includes a detailed history of Turing's work in computer science and the attempts to ground the field in formal logic."--Mathematics Teacher "This book is not for the faint hearted, as with the great masters of painting it will insist that some thought goes into appreciating it... I love the book as a book. It is a collectors item and after all what better pursuit can one have than collecting books!"--Patrick Fogarty, Mathematics TodayTable of ContentsPreface ix The Birth of Computer Science at Princeton in the 1930s Andrew W. Appel 1 Turing's Thesis Solomon Feferman 13 Notes on the Manuscript 27 Systems of Logic Based on Ordinals Alan Turing 31 A Remarkable Bibliography 141 Contributors 143

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Book SynopsisTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Chapter 1. Introduction, pg. 1*Chapter 2. Auxiliary Results, pg. 29*Chapter 3. Reduction to Restriction Estimates near the Principal Root Jet, pg. 50*Chapter 4. Restriction for Surfaces with Linear Height below 2, pg. 57*Chapter 5. Improved Estimates by Means of Airy-Type Analysis, pg. 75*Chapter 6. The Case When hlin(PHI) => 2: Preparatory Results, pg. 105*Chapter 7. How to Go beyond the Case hlin(PHI) => 5, pg. 131*Chapter 8. The Remaining Cases Where m = 2 and B = 3 or B = 4, pg. 181*Chapter 9. Proofs of Propositions 1.7 and 1.17, pg. 244*Bibliography, pg. 251*Index, pg. 257

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Book SynopsisTrade Review"An important contribution to the study of the history of mathematics, and any student, educator, or practitioner of mathematics or computer science, would benefit from reading this work."---Mark Causapin, MAA Reviews"In reading von Plato’s book the attention of the scholarly reader will be always captured."---L. Bellotti, History and Philosophy of Logic"This book presents an informed and informative hisotry of a crucially important part of mathematics. . . . a valuable addition to our corporate understanding."---Rob Ashmore, Mathematics Today

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Book SynopsisTrade Review"Fascinating... Chamberland offers enticing explanations that will leave readers hungry to know more. This wonderful book never loses its focus or momentum."--Publishers Weekly "[B]oth amateur and professional mathematicians alike will find new items of interest here... [A] welcome, splendid, fruitful addition to my math bookshelf."--Math Tango blog "The collection is outright delightful. It will agitate the minds of students and shake the sense of know-all off many a professional and most of the amateurs."--Alexander Bogomolny, Cut the Knot blog "Boring deep into the innocuous-looking number one, Chamberland opens an unexpected entry point into a dizzying maze of infinities... A bracing mathematical adventure."--Booklist "The exotics like pi and e have gotten their share of attention in the world of popular mathematical writing. Now it's time to give proper attention to the integers 1 through 9... [Single Digits] is consistently entertaining and well-written."--MAA Reviews "Chamberland takes readers on a fascinating exploration of small numbers, from one to nine, looking at their history, applications, and connections to various areas of mathematics, including number theory, geometry, chaos theory, numerical analysis, and mathematical physics... Appealing to high-school and college students, professional mathematicians, and those mesmerized by patterns, this book shows that single digits offer a plethora of possibilities that readers can count on."--DVD, Lunar and Planetary Information Bulletin "Chamberland makes this an entertaining and historical exposition, using wit and humor throughout."--Math Horizons "To put it simply, this book is a delight. Chamberland has assembled a fascinating collection of vignettes, each tied to a digit from one to nine, that inform, entertain, and intrigue... This wide spectrum of ideas is consistently interesting, and the author's skill in mining each nugget is worthy of great respect."--Choice "The range of topics included virtually guarantees that any reader will find new and unfamiliar material to enjoy... [Single Digits] is a very enjoyable book which, at many points, makes some very deep mathematics quite accessible. Highly recommended."--Keith Johnson, CMS Notes "For instructors of math courses of all levels, the vignettes in Single Digits can provide a very readable introduction or jumping-off point for discussions and projects... In an introductory group theory course, it would be a good exercise for students to consider perfect riffle shuffles in decks of size other than 52. Finally, a statistics class collecting and analyzing real-world data sets could consider whether Benford's Law applies in their situation."--Matthew Welz, MAA Focus "I highly recommend Single Digits: In Praise of Small Numbers. It would be a fine addition to any high school or math department library. As a carefully curated set of interesting topics, it would serve as a good place to start exploring the ocean of ideas in mathematics."--Bruce Cohen, NCTMTable of Contents*Frontmatter, pg. i*Contents, pg. v*Preface, pg. xi*Chapter 1. The Number One, pg. 1*Chapter 2. The Number Two, pg. 24*Chapter 3. The Number Three, pg. 69*Chapter 4. The Number Four, pg. 111*Chapter 5. The Number Five, pg. 132*Chapter 6. The Number Six, pg. 156*Chapter 7. The Number Seven, pg. 170*Chapter 8. The Number Eight, pg. 191*Chapter 9. The Number Nine, pg. 205*Chapter 10. Solutions, pg. 216*Further reading, pg. 219*Credits for illustrations, pg. 223*Index, pg. 225

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Book SynopsisTo help readers become better at solving real-world problems, this enlightening, entertaining, and inspiring book teaches simple, effective thinking techniques. The goal is not to quickly solve each challenge but to come up with as many different ways of thinking about it as possible.Trade Review"[Making Up Your Own Mind] is an elegant blend of entertainment and enlightenment."---Tom Schulte, MAA Reviews

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Book SynopsisThis book explains, in lay terms, the surprisingly simple system of mathematical logic used in digital computer circuitry. Anecdotal in its style and often funny, it follows the development of this logic system from its origins in Victorian England to its rediscovery in this century as the foundation of all modern computing machinery. ONES AND ZEROS will be enjoyed by anyone who has a general interest in science and technology.Table of ContentsBefore We Begin. Number Systems and Counting. The Basic Functions of Boolean Algebra: And, Or, And Not. Combinational Logic. The Algebra of Sets and Venn Diagrams. Other Boolean Functions. Realizing Any Boolean Function with And, Or, And Not. More Digital Circuits. Laws of Boolean Algebra. Boolean Logic. Appendix A: Counting in Base 2. Appendix B: Powers of 2. Appendix C: Summary of Boolean Functions. Further Reading. Answers to Exercises. Index. About the Author.

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Book SynopsisBerto''s highly readable and lucid guide introduces students and the interested reader to Gödel''s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel''s arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters Discusses interpretations of the Theorem made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel''s theories Written in an accessible, non-technical style Trade Review"There's Something about G¨odel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical." (Philosophia Mathematica, 2011) “There is a story that in 1930 the great mathematician John von Neumann emerged from a seminar delivered by Kurt Gödel saying: ‘It's all over.’ Gödel had just proved the two theorems about the logical foundations of mathematics that are the subject of this valuable new book by Francesco Berto. Berto's clear exposition and his strategy of dividing the proof into short, easily digestible chunks make it pleasant reading ... .Berto is lucid and witty in exposing mistaken applications of Gödel's results ... [and] has provided a thoroughly recommendable guide to Gödel's theorems and their current status within, and outside, mathematical logic.” (Times Higher Education Supplement, February 2010)Table of ContentsPrologue. Acknowledgments. Part I: The Gödelian Symphony. 1 Foundations and Paradoxes. 1 "This sentence is false". 2 The Liar and Gödel. 3 Language and metalanguage. 4 The axiomatic method, or how to get the non-obvious out of the obvious. 5 Peano's axioms … . 6 … and the unsatisfied logicists, Frege and Russell. 7 Bits of set theory. 8 The Abstraction Principle. 9 Bytes of set theory. 10 Properties, relations, functions, that is, sets again. 11 Calculating, computing, enumerating, that is, the notion of algorithm. 12 Taking numbers as sets of sets. 13 It's raining paradoxes. 14 Cantor's diagonal argument. 15 Self-reference and paradoxes. 2 Hilbert. 1 Strings of symbols. 2 "… in mathematics there is no ignorabimus". 3 Gödel on stage. 4 Our first encounter with the Incompleteness Theorem … . 5 … and some provisos. 3 Gödelization, or Say It with Numbers! 1 TNT. 2 The arithmetical axioms of TNT and the "standard model" N. 3 The Fundamental Property of formal systems. 4 The Gödel numbering … . 5 … and the arithmetization of syntax. 4 Bits of Recursive Arithmetic … . 1 Making algorithms precise. 2 Bits of recursion theory. 3 Church's Thesis. 4 The recursiveness of predicates, sets, properties, and relations. 5 … And How It Is Represented in Typographical Number Theory. 1 Introspection and representation. 2 The representability of properties, relations, and functions … . 3 … and the Gödelian loop. 6 "I Am Not Provable". 1 Proof pairs. 2 The property of being a theorem of TNT (is not recursive!) 3 Arithmetizing substitution. 4 How can a TNT sentence refer to itself? 5 γ 6 Fixed point. 7 Consistency and omega-consistency. 8 Proving G1. 9 Rosser's proof. 7 The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2. 1 G2. 2 Technical interlude. 3 "Immediate consequences" of G1 and G2. 4 Undecidable1 and undecidable2. 5 Essential incompleteness, or the syndicate of mathematicians. 6 Robinson Arithmetic. 7 How general are Gödel's results? 8 Bits of Turing machine. 9 G1 and G2 in general. 10 Unexpected fish in the formal net. 11 Supernatural numbers. 12 The culpability of the induction scheme. 13 Bits of truth (not too much of it, though). Part II: The World after Gödel. 8 Bourgeois Mathematicians! The Postmodern Interpretations. 1 What is postmodernism? 2 From Gödel to Lenin. 3 Is "Biblical proof" decidable? 4 Speaking of the totality. 5 Bourgeois teachers! 6 (Un)interesting bifurcations. 9 A Footnote to Plato. 1 Explorers in the realm of numbers. 2 The essence of a life. 3 "The philosophical prejudices of our times". 4 From Gödel to Tarski. 5 Human, too human. 10 Mathematical Faith. 1 "I'm not crazy!" 2 Qualified doubts. 3 From Gentzen to the Dialectica interpretation. 4 Mathematicians are people of faith. 11 Mind versus Computer: Gödel and Artificial Intelligence. 1 Is mind (just) a program? 2 "Seeing the truth" and "going outside the system". 3 The basic mistake. 4 In the haze of the transfinite. 5 "Know thyself": Socrates and the inexhaustibility of mathematics. 12 Gödel versus Wittgenstein and the Paraconsistent Interpretation. 1 When geniuses meet … . 2 The implausible Wittgenstein. 3 "There is no metamathematics". 4 Proof and prose. 5 The single argument. 6 But how can arithmetic be inconsistent? 7 The costs and benefits of making Wittgenstein plausible. Epilogue. References. Index.

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Book SynopsisBerto''s highly readable and lucid guide introduces students and the interested reader to Gödel''s celebrated Incompleteness Theorem, and discusses some of the most famous - and infamous - claims arising from Gödel''s arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the Theorem in separate chapters Discusses interpretations of the Theorem made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel''s theories Written in an accessible, non-technical style Trade Review"This is a beautifully clear and accurate presentation of the material, with no technical demands beyond what is required for accuracy, and filled with interesting philosophical suggestions." (John Woods, University of British Columbia) "There's Something about G¨odel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical." (Philosophia Mathematica, 2011) "There is a story that in 1930 the great mathematician John von Neumann emerged from a seminar delivered by Kurt Gödel saying: ‘It's all over.’ Gödel had just proved the two theorems about the logical foundations of mathematics that are the subject of this valuable new book by Francesco Berto. Berto's clear exposition and his strategy of dividing the proof into short, easily digestible chunks make it pleasant reading ... .Berto is lucid and witty in exposing mistaken applications of Gödel's results ... [and] has provided a thoroughly recommendable guide to Gödel's theorems and their current status within, and outside, mathematical logic.” (Times Higher Education Supplement, February 2010)Table of ContentsPrologue. Acknowledgments. Part I: The Gödelian Symphony. 1 Foundations and Paradoxes. 1 "This sentence is false". 2 The Liar and Gödel. 3 Language and metalanguage. 4 The axiomatic method, or how to get the non-obvious out of the obvious. 5 Peano's axioms … . 6 … and the unsatisfied logicists, Frege and Russell. 7 Bits of set theory. 8 The Abstraction Principle. 9 Bytes of set theory. 10 Properties, relations, functions, that is, sets again. 11 Calculating, computing, enumerating, that is, the notion of algorithm. 12 Taking numbers as sets of sets. 13 It's raining paradoxes. 14 Cantor's diagonal argument. 15 Self-reference and paradoxes. 2 Hilbert. 1 Strings of symbols. 2 "… in mathematics there is no ignorabimus". 3 Gödel on stage. 4 Our first encounter with the Incompleteness Theorem … . 5 … and some provisos. 3 Gödelization, or Say It with Numbers! 1 TNT. 2 The arithmetical axioms of TNT and the "standard model" N. 3 The Fundamental Property of formal systems. 4 The Gödel numbering … . 5 … and the arithmetization of syntax. 4 Bits of Recursive Arithmetic … . 1 Making algorithms precise. 2 Bits of recursion theory. 3 Church's Thesis. 4 The recursiveness of predicates, sets, properties, and relations. 5 … And How It Is Represented in Typographical Number Theory. 1 Introspection and representation. 2 The representability of properties, relations, and functions … . 3 … and the Gödelian loop. 6 "I Am Not Provable". 1 Proof pairs. 2 The property of being a theorem of TNT (is not recursive!) 3 Arithmetizing substitution. 4 How can a TNT sentence refer to itself? 5 γ 6 Fixed point. 7 Consistency and omega-consistency. 8 Proving G1. 9 Rosser's proof. 7 The Unprovability of Consistency and the "Immediate Consequences" of G1 and G2. 1 G2. 2 Technical interlude. 3 "Immediate consequences" of G1 and G2. 4 Undecidable1 and undecidable2. 5 Essential incompleteness, or the syndicate of mathematicians. 6 Robinson Arithmetic. 7 How general are Gödel's results? 8 Bits of Turing machine. 9 G1 and G2 in general. 10 Unexpected fish in the formal net. 11 Supernatural numbers. 12 The culpability of the induction scheme. 13 Bits of truth (not too much of it, though). Part II: The World after Gödel. 8 Bourgeois Mathematicians! The Postmodern Interpretations. 1 What is postmodernism? 2 From Gödel to Lenin. 3 Is "Biblical proof" decidable? 4 Speaking of the totality. 5 Bourgeois teachers! 6 (Un)interesting bifurcations. 9 A Footnote to Plato. 1 Explorers in the realm of numbers. 2 The essence of a life. 3 "The philosophical prejudices of our times". 4 From Gödel to Tarski. 5 Human, too human. 10 Mathematical Faith. 1 "I'm not crazy!" 2 Qualified doubts. 3 From Gentzen to the Dialectica interpretation. 4 Mathematicians are people of faith. 11 Mind versus Computer: Gödel and Artificial Intelligence. 1 Is mind (just) a program? 2 "Seeing the truth" and "going outside the system". 3 The basic mistake. 4 In the haze of the transfinite. 5 "Know thyself": Socrates and the inexhaustibility of mathematics. 12 Gödel versus Wittgenstein and the Paraconsistent Interpretation. 1 When geniuses meet … . 2 The implausible Wittgenstein. 3 "There is no metamathematics". 4 Proof and prose. 5 The single argument. 6 But how can arithmetic be inconsistent? 7 The costs and benefits of making Wittgenstein plausible. Epilogue. References. Index.

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Book SynopsisPreface.- Introduction.- Propositional Logic: Formulas, Models, Tableaux.- Propositional Logic: Deductive Systems.- Propositional Logic: Resolution.- Propositional Logic: Binary Decision Diagrams.- Propositional Logic: SAT Solvers.- First-Order Logic: Formulas, Models, Tableaux.- First-Order Logic: Deductive Systems.- First-Order Logic: Terms and Normal Forms.- First-Order Logic: Resolution.- First-Order Logic: Logic Programming.- First-Order Logic: Undecidability and Model Theory.- Temporal Logic: Formulas, Models, Tableaux.- Temporal Logic: A Deductive System.- Verification of Sequential Programs.- Verification of Concurrent Programs.- Set Theory.- Index of Symbols.- Index of Names.- Subject Index.Trade ReviewAsst. Prof. Manoj Raut, Dhirubhai Ambani Institute of Information and Communication Technology, IndiaExcerpts from full review posted Jan 15 2013 to Computing Reviews [Review #: CR140831]I have used the second edition of this book for my class. I find this new third edition more interesting and more elaborately written; I like it very much, and applaud the author for his work.Table of ContentsPreface.- Introduction.- Propositional Logic: Formulas, Models, Tableaux.- Propositional Logic: Deductive Systems.- Propositional Logic: Resolution.- Propositional Logic: Binary Decision Diagrams.- Propositional Logic: SAT Solvers.- First-Order Logic: Formulas, Models, Tableaux.- First-Order Logic: Deductive Systems.- First-Order Logic: Terms and Normal Forms.- First-Order Logic: Resolution.- First-Order Logic: Logic Programming.- First-Order Logic: Undecidability and Model Theory.- Temporal Logic: Formulas, Models, Tableaux.- Temporal Logic: A Deductive System.- Verification of Sequential Programs.- Verification of Concurrent Programs.- Set Theory.- Index of Symbols.- Index of Names.- Subject Index.

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Book SynopsisGallery of the Infinite is a mathematician's unique view of the infinitely many sizes of infinity. Written in a playful yet informative style, it introduces important concepts from set theory (including the Cantor Diagonalization Method and the Cantor-Bernstein Theorem) using colourful pictures, with little text and almost no formulas.Trade ReviewThis is a beautiful book. The pictures keep the reader engaged in a colourful mathematical journey. It is written in an engaging style suitable for over 11’s but also contains ideas that are likely to interest most adults (without the need for a refresher course, since the book does a good job of being self-contained). [...] Although a mathematician would likely be aware of many of the concepts the book presents, I would still recommend it both as a tool to intrigue others (it makes a great ‘coffee table’ book) and also since it contains many imaginative explanations and original arguments. The illustrations and narrative keep the reader entertained and make the book hard to put down." - London Mathematical Society Newsletter"This is a lovely book… Although a certain affinity with mathematical reasoning is needed the book can be read by almost anyone." - Teun Koetsier, Zentralblatt Math

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Book SynopsisThe second of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices.Table of Contents The classification of varieties Equational logic Rudiments of model theory Bibliography Index

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Book SynopsisThe third of a three-volume set of books on the theory of algebras, a study that provides a consistent framework for understanding algebraic systems, including groups, rings, modules, semigroups and lattices.Table of Contents Finite algebras and their clones Abstract clone theory Commutator theory Bibliography Index

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Book SynopsisCircular analyses of philosophical, linguistic, or computational phenomena have been attacked on the assumption that they conflict with mathematical rigour. Barwise and Moss have undertaken to prove this assumption false. This volume is concerned with extending the modelling capabilities of set theory to provide a uniform treatment of circular phenomena. As a means of guiding the reader through the concrete examples of the theory, the authors have included many exercises and solutions: these exercises range in difficulty and ultimately stimulate the reader to come up with new results. Vicious Circles is intended for use by researchers who want to use hypersets; although some experience in mathematics is necessary, the book is accessible to people with widely differing backgrounds and interests.Trade Review' ... a book to learn from.' L'Enseignement MathématiqueTable of ContentsPart I. Background: 1. Introduction; 2. Background on set theory; Part II. Vicious Circles: 3. Circularity in computer science; 4. Circularity in philosophy; 5. Circularity and paradox; Part III. Basic Theory: 6. The solution dilemma; 7. Bisimulation; Part IV. Elementary applications: 8. Graphs; 9. Modal logic; 10. Streams; 11. Games; 12. Modeling the semantic paradoxes; Part V. Further Theory: 13. Greatest fixed points; 14. Uniform operators; 15. Corecursion; Part VI. Further Applications: 16. Some applications; 17. Modeling partial information; 18. Circularity and the notion of set; 19. Conclusions and future directions.

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Book SynopsisDonald Knuth's influence in computer science ranges from the invention of literate programming to the development of the TeX programming language. One of the foremost figures in the field of mathematical sciences, his papers are widely referenced and stand as milestones of development over a wide range of topics. This volume assembles more than three dozen of Professor Knuth's pioneering contributions to discrete mathematics. It includes a variety of topics in combinatorial mathematics (finite geometries, graph theory, enumeration, partitions, tableaux, matroids, codes); discrete algebra (finite fields, groupoids, closure operators, inequalities, convolutions, Pfaffians); and concrete mathematics (recurrence relations, special numbers and notations, identities, discrete probability). Of particular interest are two fundamental papers in which the evolution of random graphs is studied by means of generating functions.Table of Contents1. Discussion of Mr. Riordan's paper 'Abel identities and inverse relations'; 2. Duality in addition chains; 3. Combinatorial analysis and computers; 4. Tables of finite fields; 5. Finite semifields and projective planes; 6. A class of projective planes; 7. Construction of a random sequence; 8. Oriented subtrees of an arc digraph; 9. Another enumeration of trees; 10. Notes on central groupoids; 11. Permutations, matrices, and generalized Young tableaux; 12. A note on solid partitions; 13. Subspaces, subsets, and partitions; 14. Enumeration of plane partitions; 15. Complements and transitive closures; 16. Permutations with nonnegative partial sums; 17. Wheels within wheels; 18. The asymptotic number of geometries; 19. Random matroids; 20. Identities from partition involutions; 21. Huffman's algorithm via algebra; 22. A permanent inequality; 23. Efficient balanced codes; 24. The power of a prime that divides a generalized binomial coefficient; 25. The first cycles in an evolving graph; 26. The birth of the giant component; 27. Polynomials involving the floor function; 28. The sandwich theorem; 29. Aztec diamonds, checkerboard graphs, and spanning trees.

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Book SynopsisSince its inception, fuzzy logic has attracted an incredible amount of interest, and this interest continues to grow at an exponential rate. As such, scientists, researchers, educators and practitioners of fuzzy logic continue to expand on the applicability of what and how fuzzy can be utilised in the real-world. In this book, the authors present key application areas where fuzzy has had significant success. The chapters cover a plethora of application domains, proving credence to the versatility and robustness of a fuzzy approach. A better understanding of fuzzy will ultimately allow for a better appreciation of fuzzy. This book provides the reader with a varied range of examples to illustrate what fuzzy logic can be capable of and how it can be applied. The text will be ideal for individuals new to the notion of fuzzy, as well as for early career academics who wish to further expand on their knowledge of fuzzy applications. The book is also suitable as a supporting text for advanced undergraduate and graduate-level modules on fuzzy logic, soft computing, and applications of AI.Table of ContentsRecognising Handwritten Digits Using a Fuzzy Neural Network Joshua Reynolds and Tianhua Chen Fuzzy Assessment of Student Academic Performances Shangen Yang and Tianhua Chen A Hybrid Fuzzy Neural Network for Image Recognition Samaresh Nayak and Tianhua Chen A Fuzzy Diagnostic System for Heart Disease Siyue Song, Tianhua Chen, and Grigoris Antoniou Analysing Medical Notes using Fuzzy Logic Siyue Song, Tianhua Chen, and Grigoris Antoniou Fostering Positive Personalisation through Fuzzy Clustering Raymond Moodley Fuzzy Logic in Modern Information Retrieval Steve Wade Fuzzy Applied to Sentiment Analysis Orestes Appel Fuzzy Logic, a Logicians Perspective Patrick Fogarty Applications of Fuzzy Logic in an Automated Warehouse Patrick Fogarty Can Fuzzy Systems Assist with Project Planning? Daniel Maia and Arjab Khuman Fuzzy Logic in Autonomous Vehicles David McDougall and Arjab Khuman AI Spawning Fuzzy Logic Fuzzy Inference System Reece Carey and Arjab Khuman The Application of Fuzzy Logic on Intelligent Transportation Systems Nath Lloyd and Arjab Khuman Fuzzy Logic Applied to Water Processes Will Chapman and Arjab Khuman Applications of Fuzzy Logic in Autonomous Vehicles Sam Asquith and Arjab Khuman Predicting Cyber Threats using Fuzzy Logic Jarrad Morden and Arjab Khuman Implementations of Fuzzy Logic in Camera Systems Sophie Hughes and Arjab Khuman Application of a Fuzzy Logic Control System for Stock Market Prediction Based on Technical Indicators and Fundamental Analysis Humza Nazir and Arjab Khuman The Application of Fuzzy Logic in Determining Outcomes of Sporting Events Spencer Deane and Arjab Khuman Using Fuzzy Logic to Educate People on Phishing Harry Taylor and Arjab Khuman

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Book SynopsisThis volume is dedicated to Hiroakira Ono life’s work on substructural logics. Chapters, written by well-established academics, cover topics related to universal algebra, algebraic logic and the Full Lambek calculus; the book includes a short biography about Hiroakira Ono. The book starts with detailed surveys on universal algebra, abstract algebraic logic, topological dualities, and connections to computer science.It further contains specialised contributions on connections to formal languages (recognizability in residuated lattices and connections to the finite embedding property), covering systems for modal substructural logics, results on the existence and disjunction properties and finally a study of conservativity of expansions. This book will be primarily of interest to researchers working in algebraic and non-classical logic.Table of ContentsChapter 1. A scientific autobiography (Hiroakira Ono).- Part I: Expository and survey chapters.- Chapter 2. Universal algebraic methods for non-classical logics (James G. Raftery).- Chapter 3. Abstract algebraic logic - An introductory chapter (Josep Maria Font).- Chapter 4. Topological duality and algebraic completions (Mai Gehrke).- Chapter 5. An algebraic glimpse at bunched implications and separation logic (Peter Jipsen and Tadeusz Litak).- Part II: Special topics.- Chapter 6. Recognizability in Residuated Lattices (José Gil-Férez and Constantine Tsinakis).- Chapter 7. Finite embeddability property for residuated lattices via regular languages (Rostislav Horčík). Chapter 8. Cover systems for the modalities of linear logic (Robert Goldblatt).- Chapter 9. A negative solution to Ono’s Problem P52: Existence and disjunction properties in intermediate predicate logic (Nobu-Yuki Suzuki).- Chapter 10. Conservative expansions of substructural logics (Jacopo Amidei, Rodolfo C. Ertola-Biraben and Franco Montagna).

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Book SynopsisThis survey of computability theory offers the techniques and tools that computer scientists (as well as mathematicians and philosophers studying the mathematical foundations of computing) need to mathematically analyze computational processes and investigate the theoretical limitations of computing. Beginning with an introduction to the mathematisation of “mechanical process” using URM programs, this textbook explains basic theory such as primitive recursive functions and predicates and sequence-coding, partial recursive functions and predicates, and loop programs. Advanced chapters cover the Ackerman function, Tarski’s theorem on the non-representability of truth, Goedel’s incompleteness and Rosser’s incompleteness theorems, two short proofs of the incompleteness theorem that are based on Lob's deliverability conditions, Church’s thesis, the second recursion theorem and applications, a provably recursive universal function for the primitive recursive functions, Oracle computations and various classes of computable functionals, the Arithmetical hierarchy, Turing reducibility and Turing degrees and the priority method, a thorough exposition of various versions of the first recursive theorem, Blum’s complexity, Hierarchies of primitive recursive functions, and a machine-independent characterisation of Cobham's feasibly computable functions.Trade Review“This textbook is suited for self-study … . As a second reading however a reader interested in rigorous proofs and/or different approaches to known concepts will benefit from this wealth of material.” (Dieter Riebesehl, zbMATH 1507.03002, 2023)Table of ContentsMathematical Background; a Review.- A Theory of Computability.- Primitive Recursive Functions.- Loop Programs.-The Ackermann Function.- (Un)Computability via Church's Thesis.- Semi-Recursiveness.- Yet another number-theoretic characterisation of P.- Godel's Incompleteness Theorem via the Halting Problem.- The Recursion Theorem.- A Universal (non-PR) Function for PR.- Enumerations of Recursive and Semi-Recursive Sets.- Creative and Productive Sets Completeness.- Relativised Computability.- POSSIBILITY: Complexity of P Functions.- Complexity of PR Functions.- Turing Machines and NP-Completeness.

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Book SynopsisThis book explores the premise that a physical theory is an interpretation of the analytico–canonical formalism. Throughout the text, the investigation stresses that classical mechanics in its Lagrangian formulation is the formal backbone of theoretical physics. The authors start from a presentation of the analytico–canonical formalism for classical mechanics, and its applications in electromagnetism, Schrödinger's quantum mechanics, and field theories such as general relativity and gauge field theories, up to the Higgs mechanism.The analysis uses the main criterion used by physicists for a theory: to formulate a physical theory we write down a Lagrangian for it. A physical theory is a particular instance of the Lagrangian functional. So, there is already an unified physical theory. One only has to specify the corresponding Lagrangian (or Lagrangian density); the dynamical equations are the associated Euler–Lagrange equations. The theory of Suppes predicates as the main tool in the axiomatization and examples from the usual theories in physics. For applications, a whole plethora of results from logic that lead to interesting, and sometimes unexpected, consequences.This volume looks at where our physics happen and which mathematical universe we require for the description of our concrete physical events. It also explores if we use the constructive universe or if we need set–theoretically generic spacetimes.Trade Review“This book is a compilation, ‘an essay’, of the bulk of their work from 1990 to the present. This 191 page essay includes some historical background and lots of snippets and parts of da Costa and Doria’s work on the meta-mathematics of mathematical physics. It starts with a primer on graduate-level basic physics … ending with a consideration of hypercomputation.” (Deborah Konkowski, zbMATH 1494.00005, 2022)Table of ContentsForeword1. PreliminaryPart I. Physics: A Primer2. Classical mechanics3. Variational calculus4. Lagrangian formulation5. Hamilton’s equations6. Hamilton–Jacobi theory7. Where the action is8. From classical to quantum9. Field theory10. Electromagnetism11. Special relativity12. General relativity13. Gauge field theoriesPart II. Axiomatics14. Axiomatizations in ZFCPart III. Technicalities15. HierarchiesPart IV. More applications16. Arnol’d’s 1974 problems17. Forcing and gravitation18. Economics and ecology.Part V. Computer science19. Fast–growing functionsPart VI. Hypercomputation20. HypercomputationReferences

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Book SynopsisThis textbook covers topics of undergraduate mathematics in abstract algebra, geometry, topology and analysis with the purpose of connecting the underpinning key ideas. It guides STEM students towards developing knowledge and skills to enrich their scientific education. In doing so it avoids the common mechanical approach to problem-solving based on the repetitive application of dry formulas. The presentation preserves the mathematical rigour throughout and still stays accessible to undergraduates. The didactical focus is threaded through the assortment of subjects and reflects in the book’s structure.Part 1 introduces the mathematical language and its rules together with the basic building blocks. Part 2 discusses the number systems of common practice, while the backgrounds needed to solve equations and inequalities are developed in Part 3. Part 4 breaks down the traditional, outdated barriers between areas, exploring in particular the interplay between algebra and geometry. Two appendices form Part 5: the Greek etymology of frequent terms and a list of mathematicians mentioned in the book. Abundant examples and exercises are disseminated along the text to boost the learning process and allow for independent work.Students will find invaluable material to shepherd them through the first years of an undergraduate course, or to complement previously learnt subject matters. Teachers may pick’n’mix the contents for planning lecture courses or supplementing their classes.Trade Review“The book being reviewed is a collection of what the author considers to be essential material for undergraduates … . it has to be said that many students will find that there is plenty to learn from this well-written book, which would also be a useful reference text had there been a properly compiled index.” (Peter Shiu, The Mathematical Gazette, Vol. 107 (570), November, 2023)Table of ContentsPart I: Basic Objects and Formalisation - Round-up of Elementary Logic.- Naive Set Theory.- Functions.- More Set Theory and Logic.- Boolean Algebras. Part 2: Numbers and Structures - Intuitive Arithmetics.- Real Numbers.- Totally Ordered Spaces.- Part 3: Elementary Real Functions - Real Polynomials.- Real Functions of One Real Variables.- Algebraic Functions.- Elementary Transcendental Functions.- Complex Numbers.- Enumerative Combinatorics.- Part 4: Geometry through Algebra - Vector Spaces.- Orthogonal Operators.- Actions & Representations.- Elementary Plane Geometry.- Metric Spaces.- Part 5: Appendices - Etymologies.- Index of names.- Main figures.- Glossary.- References.

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Book SynopsisThis open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems.

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Book SynopsisThis volume explores A.P. Morse’s (1911-1984) development of a formal language for writing mathematics, his application of that language in set theory and mathematical analysis, and his unique perspective on mathematics. The editor brings together a variety of Morse’s works in this compilation, including Morse's book A Theory of Sets, Second Edition (1986), in addition to material from another of Morse’s publications, Web Derivatives, and notes for a course on analysis from the early 1950's. Because Morse provided very little in the way of explanation in his written works, the editor’s commentary serves to outline Morse’s goals, give informal explanations of Morse’s formal language, and compare Morse’s often unique approaches to more traditional approaches. Minor corrections to Morse’s previously published works have also been incorporated into the text, including some updated axioms, theorems, and definitions. The editor’s introduction thoroughly details the corrections and changes made and provides readers with valuable insight on Morse’s methods.A.P. Morse’s Set Theory and Analysis will appeal to graduate students and researchers interested in set theory and analysis who also have an interest in logic. Readers with a particular interest in Morse’s unique perspective and in the history of mathematics will also find this book to be of interest.Table of ContentsPreface.- Editor's Introduction.- Language and Inference.- Logic.- Set Theory.- Elementary Analysis.- Metrics.- Measure.- Linear Measure and Total Variation.- Integration.- Product Measures.- Web Derivatives.- Classical Differentiation.- The Construction of Definition.- The Consistency of the Axiom of Size.- Suggested Reading.- Publications of A.P. Morse.- Errata to A Theory of Sets, Second Edition.- Integration with Respect to Addor Functions.- The Henstock-Kurzweil Integral.

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Book SynopsisThis book constitutes revised selected papers from the refereed proceedings of the 4th International Workshop on Dynamic Logic, DaLí 2022, held in Haifa, Israel, in July/August 2022.The 8 full papers presented in this volume were carefully reviewed and selected from 22 submissions. They deal with new trends and applications in the area of Dynamic Logic. Table of ContentsFirst steps in updating knowing how.- Parametrized modal logic II: the unidimensional case.- Relating Kleene algebras.- Dynamic epistemic logic for budget-constrained agents.- Action models for coalition logic.- Quantum logic for observation of physical quantities.- Cautious distributed belief.- A STIT logic of intentionality.

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Book SynopsisEdited in collaboration with FoLLI, this book constitutes the refereed proceedings of the 10th Indian Conference on Logic and Its Applications, ICLA 2023, which was held in Indore, India, in March 2023.Besides 6 invited papers presented in this volume, there are 9 contributed full papers which were carefully reviewed and selected from 18 submissions. The volume covers a wide range of topics. These topics are related to modal and temporal logics, intuitionistic connexive and imperative logics, systems for reasoning with vagueness and rough concepts, topological quasi-Boolean logic and quasi-Boolean based rough set models, and first-order definability of path functions of graphs.Table of ContentsA Note on the Ontology of Mathematics.- Boolean Functional Synthesis: From Under the Hood of Solvers.- Labelled Calculi for Lattice-based Modal Logics.- Two Ways to Scare a Gruffalo.- Determinacy Axioms and Large Cardinals.- Big ideas from logic for mathematics and computing education.- Modal Logic of Generalized Separated Topological Spaces.- Multiple-valued Semantics for Metric Temporal Logic.- Segment transit function of the induced path function of graphs and its first-order definability.- Fuzzy Free Logic with Dual Domain Semantics.- A New Dimension of Imperative Logic. -Quasi-Boolean based models in Rough Set theory: A case of Covering.- Labelled calculi for the logics of rough concepts.- An Infinity of Intuitionistic Connexive Logics.- Relational Semantics for Normal Topological Quasi-Boolean Logic.

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Book SynopsisThis book constitutes the proceedings of the 5th International Workshop on Formal Methods Teaching, FMTea 2023, which was held in Lübeck, Germany, in March 2023.The 7 full papers presented in this volume were carefully reviewed and selected from 10 submissions. FMTea 2023 aim is to support a worldwide improvement in learning Formal Methods, mainly by teaching but also via self-learning.Table of ContentsAutomated Exercise Generation for Satisfiability Checking.- Graphical Loop Invariant Based Programming.- A Gentle Introduction to Verification of Parameterized Reactive Systems.- Model Checking Concurrent Programs for Autograding in pseuCo Book.- Teaching TLA+ to Engineers at Microsoft.- Teaching and Training in Formalisation with B.- Teaching low-code Formal Methods with Coloured Petri Nets.

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Book SynopsisFrom the ‘punctuated equilibrium' of Eldrege and Gould, through Lewontin's ‘triple helix' and the various visions and revisions of the Extended Evolutionary Synthesis (EES) of Laland and others, both data and theory have demanded an opening-up of the 1950's Evolutionary Synthesis that so firmly wedded evolutionary theory to the mathematics of gene frequency analysis. It can, however, be argued that a single deep and comprehensive mathematical theory may simply not be possible for the almost infinite varieties of evolutionary process active at and across the full range of scales of biological, social, institutional, and cultural phenomena. Indeed, the case history of 'meme theory' should have raised a red flag that narrow gene-centered models of evolutionary process may indeed have serious limitations. What is attempted here is less grand, but still broader than a gene-centered analysis. Following the instruction of Maturana and Varela that all living systems are cognitive, in a certain sense, and that living as a process is a process of cognition, the asymptotic limit theorems of information and control theories that bound all cognition provide a basis for constructing an only modestly deep but wider-ranging series of probability models that might be converted into useful statistical tools for the analysis of observational and experimental data related to evolutionary process. The line of argument in this series of interrelated essays proves to be surprisingly direct.Table of Contents1 Onthemajortransitions1.1 Introduction1.2 Symmetryandsymmetry-breaking1.3 Resources1.4 Cognitioninnonergodicsystems1.5 Theprebiotic`bigbang'1.6 Biological`recombinationtransparency'1.7 Asimpleapplication1.8 Specializationandcooperation:multipleworkspaces1.9 Discussion1. MathematicalAppendix1. References2 OntheExtendedEvolutionarySynthesis2.1 Introduction2.2 Firstnotions2.3 Thebasictheory2.4 Examples2.5 Moretheory:selectionpressureasshadowprice2.6 Extendingthemodels2.7 Discussion2.8 MathematicalAppendix2.9 References3O nregulation3.1 Introduction3.2 Theory3.3 Applications3.4 Discussion3.5 MathematicalAppendix3.6 References4 Punctuatedregulationasanevolutionarymechanism4.1 Introduction4.2 FisherZerosreconsidered4.3 ExtinctionI:Simplenoise-inducedtransitions4.4 ExtinctionII:Morecomplicatednoise-inducedtransitions4.5 ExtinctionIII:Environmentalshadowprice4.6 Discussion4.7 MathematicalAppendix4.8 References5 Institutionaldynamicsunderselectionpressureanduncertainty5.1 Introduction5.2 ARateDistortionTheoremmodelofcontrol5.3 Selectionpressuredynamics5.4 Destabilizationbydelay5.5 ExtendingtheDataRateTheorem5.6 Movingon5.7 Reconsideringcognition\textit{AnSich5.8 Changingtheviewpoint5.9 Discussion5. References6O n`Speciation':Fragmentsizeininformationsystemphasetransitions6.1 Introduction6.2`Simple'phasetransition6.3 Phasetransitionsinnetworksofinformation-exchangemodules6.4 Discussion6.5 MathematicalAppendix:`Biological'renormalizations6.6 References7 Adaptingcognitionmodelstobiomolecularcondensatedynamics7.1 Introduction7.2 Resources7.3 Cognition7.4 PhasetransitionsI:Fisherzeros7.5 Cognitive`reactionrate'7.6 PhasetransitionsII:Signaltransductionandnoise7.7 Discussion7.8 MathematicalAppendix:Groupoids7.9 References8 EvolutionaryExaptation:Sharedinterbrainactivityinsocialcommunication8.1 Introduction8.2 Correlation8.3 Cognition8.4 Dynamics8.5 Cognitionrate8.6 Anexample8.7 Cooperation:Multipleworkspaces8.8 Networktopologyisimportant8.9 Timeandresourceconstraintsareimportant8.10 Furthertheoreticaldevelopment8.11 Discussion8.12 MathematicalAppendix8.13 References9 Afterward

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Book SynopsisThe starting point for this monograph is the previously unknown connection between the Continuum Hypothesis and the saturation of the non-stationary ideal on ω1; and the principle result of this monograph is the identification of a canonical model in which the Continuum Hypothesis is false. This is the first example of such a model and moreover the model can be characterized in terms of maximality principles concerning the universal-existential theory of all sets of countable ordinals. This model is arguably the long sought goal of the study of forcing axioms and iterated forcing but is obtained by completely different methods, for example no theory of iterated forcing whatsoever is required. The construction of the model reveals a powerful technique for obtaining independence results regarding the combinatorics of the continuum, yielding a number of results which have yet to be obtained by any other method. This monograph is directed to researchers and advanced graduate students in Set Theory. The second edition is updated to take into account some of the developments in the decade since the first edition appeared, this includes a revised discussion of Ω-logic and related matters.

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Book SynopsisThis book analyzes the generation of the arrow-categories of a given category, which is a foundational and distinguishable Category Theory phenomena, in analogy to the foundational role of sets in the traditional set-based Mathematics, for defi nition of natural numbers as well. This inductive transformation of a category into the infinite hierarchy of the arrowcategories is extended to the functors and natural transformations. The author considers invariant categorial properties (the symmetries) under such inductive transformations. The book focuses in particular on Global symmetry (invariance of adjunctions) and Internal symmetries between arrows and objects in a category (in analogy to Field Theories like Quantum Mechanics and General Relativity). The second part of the book is dedicated to more advanced applications of Internal symmetry to Computer Science: for Intuitionistic Logic, Untyped Lambda Calculus with Fixpoint Operators, Labeled Transition Systems in Process Algebras and Modal logics as well as Data Integration Theory.

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Book SynopsisThe two main themes of this book, logic and complexity, are both essential for understanding the main problems about the foundations of mathematics. Logical Foundations of Mathematics and Computational Complexity covers a broad spectrum of results in logic and set theory that are relevant to the foundations, as well as the results in computational complexity and the interdisciplinary area of proof complexity. The author presents his ideas on how these areas are connected, what are the most fundamental problems and how they should be approached. In particular, he argues that complexity is as important for foundations as are the more traditional concepts of computability and provability.Emphasis is on explaining the essence of concepts and the ideas of proofs, rather than presenting precise formal statements and full proofs. Each section starts with concepts and results easily explained, and gradually proceeds to more difficult ones. The notes after each section present some formal definitions, theorems and proofs.Logical Foundations of Mathematics and Computational Complexity is aimed at graduate students of all fields of mathematics who are interested in logic, complexity and foundations. It will also be of interest for both physicists and philosophers who are curious to learn the basics of logic and complexity theory.Trade Review“This monograph by the outstanding Czech logician Pavel Pudlák provides a broad but also deep survey of work in logic and computer science relevant to foundational issues, interpreted in a wide sense. … This is a fine overview of logic and complexity theory that can be confidently recommended to anybody who would like to orient themselves in an increasingly intricate and difficult field.” (Alasdair Urquhart, Philosophia Mathematica, Vol. 23 (3), October, 2015)“For the non-expert it offers indeed a ‘gentle introduction’ to logic that is well selected and excellently explained. And for the logician it certainly offers some of the best introductions to those topics outside their area of direct expertise. … it contains plenty of informal explanations, intuition and motivation. … It is truly a gift to the logic and wider communities … . This book is very enjoyable to read and I wish it all success.” (Olaf Beyersdorff, Mathematical Reviews, August, 2014)“It spans the historical, logical, and at times philosophical underpinnings of the theory of computational complexity. Students of mathematics seeking a transition to higher mathematics will find it helpful, as will mathematicians with expertise in other areas. … an excellent choice for a first text in studying complexity, or as a clarifying adjunct to any assigned text in this area. … a compact guide for graduate students with a need for or interest in computational complexity and its foundations.” (Tom Schulte, MAA Reviews, July, 2014)“This book, exactly as indicated by its title, deals with the main philosophical, historical, logical and mathematical aspects … in a quite approachable and attractive way. … the prospective readers of this book are mathematicians with an interest in the foundations, philosophers with a good background in mathematics, and also philosophically minded scientists. Due to the author’s nice style, the book will be a very good choice for the first text in studying this subject.” (Branislav Boričić, zbMATH, Vol. 1270, 2013)Table of Contents​​​​​​​​​​Mathematician’s world.- Language, logic and computations.- Set theory.- Proofs of impossibility.- The complexity of computations.- Proof complexity.- Consistency, Truth and Existence.- References.

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Book SynopsisGerhard Gentzen is best known for his development of the proof systems of natural deduction and sequent calculus, central in many areas of logic and computer science today. Another noteworthy achievement is his resolution of the embarrassing situation created by Gödel's incompleteness results, especially the second one about the unprovability of consistency of elementary arithmetic. After these successes, Gentzen dedicated the rest of his short life to the main problem of Hilbert's proof theory, the question of the consistency of analysis. He was arrested in the summer of 1945 with other professors of the German University of Prague and died soon afterward of starvation in a prison cell. Attempts at locating his lost manuscripts failed at the time, but several decades later, two slim folders of shorthand notes were found. In this volume, Jan von Plato gives an overview of Gentzen's life and scientific achievements, based on detailed archival and systematic studies, and essential for placing the translations of shorthand manuscripts that follow in the right setting. The materials in this book are singular in the way they show the birth and development of Gentzen's central ideas and results, sometimes in a well-developed form, and other times as flashes into the anatomy of the workings of a unique mind.Trade Review“This book is obviously indispensable to historians of logic in the immediate wake of Gödel’s 1931 incompleteness theorems. … Saved from the Cellar is also valuable for less specialist readers (like myself ) who wish to understand the broader outlines of what proof theory has meant to several of its leading creators.” (Colin McLarty, Isis, Vol. 111 (1), 2020)“The book contains translations of shorthand notes which survived in the Nachlass of the mathematical logician Gerhard Gentzen. ... The book is valuable source for the history of modern logic; the editor did an excellent work in getting the shorthand notes, first transcribed in normal German text, and then translating it to English.” (Reinhard Kahle, zbMath 1414.03002, 2019)“Every general reader interested in modern logic and its history, … may find a source of inspiration in Genzen’s unpublished notes of the thirties, as well as for the philosopher concerned with epistemological aspects of modern logic.” (Adrian Rezus, Studia Logica, Vol. 107, 2019)“This is an account and transcription of two slim folders of stenographic material in Gerhard Gentzen's handwriting that were found in 1984. … this book is a valuable contribution to the history of the development of mathematical logic in the first half of the twentieth century.” (Henry Africk, Mathematical Reviews, December, 2017)Table of ContentsPart I: A Sketch of Gentzen's Life and Work.- 1. Overture.- 2. Gentzen's years of study.- Dr. Gentzen's arduous years in Nazi Germany.- 4. The scientific accomplishments.- 5. Loose ends.- 6. Gentzen's genuis.- Part II: Overview of the Shorthand Notes.- 1. Gentzen's series of stenographic manuscripts.- 2. The items in this collection.- Practical remarks on the manuscripts.- Manuscript illustrations.- The German alphabet in Latin, Sutterlin, and Fraktur Type.- Bibliography for parts I and II.- Index of names for Parts I and II.- Part III: The Original Writings.- 1. Reduction of number-theoretic problems to predicate logic.- 2. Replacement of functions by predicates.- 3. The formation of abstract concepts.- 4. Five different forms of natural calculi.- 5. Formal conception of correctness in arithmetic I.- 6. Investigations into logical inferences.- 7. Reduction of classical to intuitionistic logic.- 8. CV of the candidate Gerhard Gentzen.-0 9. Letters to Heyting.- 10. Formal conception of correctness in arithmetic II.- 11. Proof theory of number theory.- 12. Consistency of artihmetic, for publication.- 13. Correspondence with Paul Bernays.- 14. Forms of type theory.- 15. Predicate logic.- 16. Propositional logic.- 17. Foundational research in mathematics.- Table of cross-references in the Gentzen papers.- Index of names in the Gentzen papers.- Index of subjects in the Gentzen papers.

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Book SynopsisDas Buch setzt sich zum Ziel, auch mathematisch wenig vorgebildete Leser in die klassische zweiwertige Logik und ihre intensionalen Erweiterungen wie Modal-Logik, Zeit-Logik und dynamische Logik einzuführen. Die hier näher betrachteten intensionalen Systeme hängen zusammen mit Fragen aus der Beweistheorie der Peano-Arithmetik, Korrektheitsfragen in der Theorie der Programmiersprachen und mit Problemen, die die Semantik natürlicher Sprachen betreffen.Table of ContentsAussagenlogik - modale Aussagenlogik und Varianten - Grundbegriffe der Prädikatenlogik - Herbrandscher Satz - Gödelscher Vollständigkeitssatz - modale Aspekte der Gödelschen Unvollständigkeitssätze - modelltheoretische Begriffe - modale und dynamische Prädikatenlogik - höherstufige Prädikatenlogik und Typentheorie.

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Book SynopsisLectures on intuitionism.- Realizability: A retrospective survey.- Some applications of Kleene's methods for intuitionistic systems.- Notes on intuitionistic second order arithmetic.- Some properties of intuitionistic zermelo-frankel set theory.- Ouelques Resultats sur les Interpretations Fonctionnelles.- Combinator realizability of constructive finite type analysis.- The arithmetic theory of constructions.- The priority method for the construction of recursively enumerable sets.- Admissible ordinals and priority arguments.- Abstract computability versus analog-generability (a survey).- Infinitary combinatorics.- The maximum sum of a family of ordinals.- Effective implications between the "finite" choice axioms.- On descendingly complete ultrafilters.- XVI. A model for the negation of the axiom of choice.- Filters closed under MAHLO's and GAIFMAN's operation.- On chromatic number of graphs and set systems.- Countable models of set theories.- Errata.- Descriptive set theory in .- Modal model theory.- A preservation theorem for interpretations.- Vaught sentences and Lindström's regular relations.Table of ContentsLectures on intuitionism.- Realizability: A retrospective survey.- Some applications of Kleene's methods for intuitionistic systems.- Notes on intuitionistic second order arithmetic.- Some properties of intuitionistic zermelo-frankel set theory.- Ouelques Resultats sur les Interpretations Fonctionnelles.- Combinator realizability of constructive finite type analysis.- The arithmetic theory of constructions.- The priority method for the construction of recursively enumerable sets.- Admissible ordinals and priority arguments.- Abstract computability versus analog-generability (a survey).- Infinitary combinatorics.- The maximum sum of a family of ordinals.- Effective implications between the "finite" choice axioms.- On descendingly complete ultrafilters.- XVI. A model for the negation of the axiom of choice.- Filters closed under MAHLO's and GAIFMAN's operation.- On chromatic number of graphs and set systems.- Countable models of set theories.- Errata.- Descriptive set theory in .- Modal model theory.- A preservation theorem for interpretations.- Vaught sentences and Lindström's regular relations.

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Book SynopsisThe territory of preserver problems has grown continuously within linear analysis. This book presents a cross-section of the modern theory of preservers on infinite dimensional spaces (operator spaces and function spaces) through the author's corresponding results. Special emphasis is placed on preserver problems concerning some structures of Hilbert space operators which appear in quantum mechanics. In addition, local automorphisms and local isometries of operator algebras and function algebras are discussed in detail.Trade ReviewFrom the reviews: "Preserver problems deal with maps on subsets of algebras that preserve certain sets, relations, functions, etc. … The book is very well organized. … One of its remarkable features is that it links several areas, particularly operator theory and mathematical physics. The audience of this book is therefore potentially wide: operator algebraists, mathematical physicists, linear algebraists, ring theorists, etc. I warmly recommend this book to anyone interested in preserver problems." (Matej Brešar, Mathematical Reviews, Issue, 2007 g) "The monograph under review collects many important and highly nontrivial results and efforts. It is important to recall that the basic material is based on the research done by the author, who belongs to the eminent researchers in this field. The style is very fresh … . I recommend to book for students and experts interested in operator algebra, noncommutative measure theory and mathematical foundations of quantum physics. The monograph is welcome in the quantum structures realm." (Anatolij Dvurecenskij, Zentralblatt MATH, Vol. 1119 (21), 2007)Table of ContentsSome Linear and Multiplicative Preserver Problems on Operator Algebras and Function Algebras.- Preservers on Quantum Structures.- Local Automorphisms and Local Isometries of Operator Algebras and Function Algebras.

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Book SynopsisOver the years, this book has become a standard reference and guide in the set theory community. It provides a comprehensive account of the theory of large cardinals from its beginnings and some of the direct outgrowths leading to the frontiers of contemporary research, with open questions and speculations throughout.Table of ContentsPreliminaries.- Beginnings.- Partition Properties.- Forcing and Sets of Reals.- Aspects of Measurability.- Strong Hypotheses.- Determinacy.

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Book SynopsisDer Band 1A beginnt mit einem Vorwort zur Gesamtedition. Den Hauptteil des Bandes bilden Hausdorffs Arbeiten über geordnete Mengen aus den Jahren 1901-1909. Diese haben der Entwicklung der Mengenlehre nachhaltige Impulse verliehen. Sie enthalten zahlreiche für die Untersuchung geordneter Mengen grundlegende neue Begriffe sowie tiefliegendere Resultate. Alle diese Arbeiten sind sorgfältig kommentiert. Die Kommentare zeigen, dass einige von Hausdorff's Ideen und Resultaten für die moderne Grundlagenforschung hochaktuell sind. Ferner enthält der Band Hausdorff's kritische Besprechung von Russells "The Principles of Mathematics", aus dem Nachlass seine Vorlesung "Mengenlehre" von 1901 (eine der ersten Vorlesungen über dieses Gebiet überhaupt) sowie einen Essay "Hausdorff als akademischer Lehrer". Table of ContentsTeil I. Arbeiten über geordnete Mengen.– Über eine gewisse Art geordneter Mengen.- Kommentar.- Der Potenzbegriff in der Mengenlehre.- Kommentar.- Untersuchungen über Ordnungstypen I, II, III.- Untersuchungen über Ordnungstypen IV, V.- Kommentar.- Über dichte Ordnungstypen.- Kommentar.- Grundzüge einer Theorie der geordneten Mengen.- Kommentar.- Die Graduierung nach dem Endverlauf.- Comments.- Summe von N1 Mengen.- Comments.- Gaps in partially ordered sets and related problems.- Teil II. Aus dem Nachlaß zur Mengenlehre.- Mengenlehre. Vorlesung der Universität Leipzig, Sommersemester 1901.- Kommentar.- Alefsätze.- Anhänge.- Bertrand Russell, The Principles of Mathematics (Besprechung).- Kommentar.- Hausdorff als akademischer Lehrer.- Entstehung der Hausdorff-Edition.- Personenregister.- Sachregister.

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Book SynopsisIm Jahr 1931 erschien im Monatsheft für Mathematik und Physik ein Artikel mit dem geheimnisvoll klingenden Titel Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. In dieser Arbeit hat Kurt Gödel zwei Unvollständigkeitssätze bewiesen, die unseren Blick auf die Mathematik von Grund auf verändert haben. Gödels Sätze manifestieren, dass zwischen dem Begriff der Wahrheit und dem Begriff der Beweisbarkeit eine Kluft besteht, die wir nicht überwinden können. Die Mathematik fügt sich in kein formales Korsett.Seit ihrer Entdeckung sind die Unvollständigkeitssätze in aller Munde und eine Flut an Büchern widmet sich ihrem fulminanten Inhalt. Doch kaum ein Werk behandelt die Gödel‘sche Arbeit in ihrer ursprünglichen Form − und dies hat triftige Gründe: Seine komplexen, in akribischer Präzision beschriebenen Argumentationsketten, die vielen Definitionen und Sätze und die heute weitgehend überholte Notation machen Gödels historisches Meisterwerk zu einer schwer zu lesenden Arbeit.In diesem Buch wird Gödels Beweis aus dem Jahr 1931 detailliert aufgearbeitet. Alle Einzelschritte werden erläutert und anhand zahlreicher Beispiele verständlich erklärt. Doch dieses Buch ist mehr als eine kommentierte Fassung der historischen Arbeit. Die Beweise der Unvollständigkeitssätze in vollem Umfang zu verstehen, bedingt, die Geschichte zu verstehen, und so versetzen zahlreiche Exkurse den Leser in die Zeit zu Beginn des zwanzigsten Jahrhunderts zurück. Es ist die Zeit, in der die Mathematik die größte Krise ihrer Geschichte durchlebte, die Typentheorie und die axiomatische Mengenlehre Gestalt annahmen und sich Hilberts formalistische Logik und Brouwers intuitionistische Mathematik mit offenem Visier gegenüber standen.Die 2. Auflage ist vollständig durchgesehen.Stimme zur ersten Auflage: „...eine didaktisch sehr gut gemachte Darstellung.“ Prof. Dr. Matthias Homeister, FH BrandenburgTable of ContentsEinleitung.- Die formalen Grundlagen der Mathematik.- Beweisskizze.- Das System P.- Primitiv-rekursive Funktionen.- Die Grenzen der Mathematik.- Epilog.

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Book SynopsisDieses umfassende Lehrbuch wurde geschrieben für Studenten und Dozenten der Mathematik und Informatik, und wegen der ausführlichen Darstellung der Gödelschen Unvollständigkeitssätze auch für Fachstudenten der Philosophischen Logik. Für diese Neuauflage wurde der Text sachlich und stilistisch vollständig überarbeitet, er enthält verbesserte Beweise und Übungen mit Lösungshinweisen sowie eine historisch orientierte Einleitung. Das Buch kann ganz unabhängig von Vorlesungen aber auch zum Selbststudium genutzt werden. Table of ContentsAussagenlogik - Prädikatenlogik - Syntax und Semantik - Der Gödelsche Vollständigkeitssatz - Nichtstandardmodelle - Logikprogammierung - Resolution und Unifikation - Elemente der Modelltheorie - Ehrenfeucht-Spiele und Ultraprodukte - Entscheidbarkeit, Unentscheidbarkeit und Unvollständigkeit - Lösungshinweise zu den Übungen

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