Mathematical logic Books

Book SynopsisThis textbook addresses the mathematical description of sets, categories, topologies and measures, as part of the basis for advanced areas in theoretical computer science like semantics, programming languages, probabilistic process algebras, modal and dynamic logics and Markov transition systems. Using motivations, rigorous definitions, proofs and various examples, the author systematically introduces the Axiom of Choice, explains Banach-Mazur games and the Axiom of Determinacy, discusses the basic constructions of sets and the interplay of coalgebras and Kripke models for modal logics with an emphasis on Kleisli categories, monads and probabilistic systems. The text further shows various ways of defining topologies, building on selected topics like uniform spaces, Gödel’s Completeness Theorem and topological systems. Finally, measurability, general integration, Borel sets and measures on Polish spaces, as well as the coalgebraic side of Markov transition kernels along with applications to probabilistic interpretations of modal logics are presented. Special emphasis is given to the integration of (co-)algebraic and measure-theoretic structures, a fairly new and exciting field, which is demonstrated through the interpretation of game logics. Readers familiar with basic mathematical structures like groups, Boolean algebras and elementary calculus including mathematical induction will discover a wealth of useful research tools. Throughout the book, exercises offer additional information, and case studies give examples of how the techniques can be applied in diverse areas of theoretical computer science and logics. References to the relevant mathematical literature enable the reader to find the original works and classical treatises, while the bibliographic notes at the end of each chapter provide further insights and discussions of alternative approaches.Trade Review“The book is absolutely packed with ideas and theorems, classical results, well-knowns as well as obscure pieces of mathematics, familiar and less familiar axioms, and generally many many facts, ranging from trivial to deep and pivotal. … book is probably the most original I’ve seen in terms of its contents. … ideas and presentation is clearly derived and dictated by the author’s renowned expertise in the area, producing a very unique point of view on an active area of research.” (Ittay Weiss, MAA Reviews, maa.org, April, 2017)“This volume is targeted at giving sound mathematical foundations for advanced topics in theoretical computer science … . The author always gives accurate definitions, complete proofs and lots of different examples clarifying and motivating the concepts. Each chapter has lots of exercises sometimes pointing to further developments. … there is a comprehensive list of references to examples and counterexamples for various propositions and assertions given in the text. The style is always motivating and sometimes humorous.” (Dieter Riebesehl, zbMATH 1334.68002, 2016)Table of ContentsPreface.- 1 The Axiom of Choice and Some of Its Equivalents.- 2 Categories.- 3 Topological Spaces.- 4 Measures for Probabilistic Systems.- List of Examples.- References.- Index.

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Book SynopsisThis series of lectures delivered by Lonergan at Boston College in 1957 deal with the general character of mathematical logic and its relation to truth, Scholasticism, and Aristotelian logic.

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Book SynopsisBoolean Rings.- Boolean Algebras.- Boolean Algebras Versus Rings.- The Principle of Duality.- Fields of Sets.- Elementary Relations.- Order.- Infinite Operations.- Topology.- Regular Open Sets.- Subalgebras.- Homomorphisms.- Extensions of Homomorphisms.- Atoms.- Finite Boolean Algebras.- Atomless Boolean Algebras.- Congruences and Quotients.- Ideals and Filters.- Lattices of Ideals.- Maximal Ideals.- Homomorphism and Isomorphism Theorems.- The Representation Theorem.- Canonical Extensions.- Complete Homomorphisms and Complete Ideals.- Completions.- Products of Algebras.- Isomorphisms of Factors.- Free Algebras.- Boolean s-algebras.- The Countable Chain Condition.- Measure Algebras.- Boolean Spaces.- Continuous Functions.- Boolean Algebras and Boolean Spaces.- Duality for Ideals.- Duality for Homomorphisms.- Duality for Subalgebras.- Duality for Completeness.- Boolean s-spaces.- The Representation of s-algebras.- Boolean Measure Spaces.- Incomplete Algebras.- Duality for Products.- SumsTrade ReviewFrom the reviews:“This is an excellent and much-needed comprehensive undergraduate textbook on Boolean algebras. It contains a complete and thorough introduction to the fundamental theory of Boolean algebras. Aimed at undergraduate mathematics students, the book is, in the first author’s words, “a substantially revised version of Paul Halmos’ “Lectures on Boolean Algebras.” It certainly achieves its stated goal of “steering a middle course between the elementary arithmetic aspects of the subject” and “the deeper mathematical aspects of the theory” of Boolean algebras.”…“The book is written for undergraduate students who already have skills in proving theorems. However, since the proofs are so detailed and clear, it could work well as a text for a second or even first course involving substantial proofs. For this reason, it would also make a great book for a student doing independent study. The text is somewhat informal in the sense that sometimes proofs appear in the prose rather than under the heading, “Proof”, but it is always clear when this is being done. Though the book starts with an introduction to Boolean rings, knowledge of group theory or rings is not a prerequisite for using the book.”… “In summary, “Introduction to Boolean algebras” is a gem of a text which fills a long-standing gap in the undergraduate literature. It combines the best of both worlds by rigorously covering all the fundamental theorems and topics of Boolean algebra while at the same time being easy to read, detailed, and well-paced for undergraduate students. It is my most highly recommended text for undergraduates studying Boolean algebras.”(Natasha Dobrinen. The Bulletin of Symbolic Logic, Vol. 16 (2), June 2010: 281-282)"Introduction to Boolean Algebras … is intended for advanced undergraduates. Givant (Mills College) and Halmos … using clear and precise prose, build the abstract theory of Boolean rings and algebras from scratch. … the necessary topological material is developed within the book and an appendix on set theory is included. … Includes an extensive bibliography and more than 800 exercises at all levels of difficulty. Summing Up: Highly recommended. Upper-division undergraduates, graduate students, researchers, and faculty." (S. J. Colley, Choice, Vol. 46 (10), June 2009)“The authors have written a book for advanced undergraduates and beginning graduate students. … The authors start with the definition of Boolean rings and Boolean algebras, give examples and basic facts and compare both notions. … There are a large number of exercises of varying level of difficulty. Hints for the solutions of the harder problems are given in an appendix. A detailed solutions manual for all exercises is available for instructors. The book can serve as a basis for a variety of courses.” (Martin Weese, Zentralblatt MATH, Vol. 1168, 2009)Table of ContentsBoolean Rings.- Boolean Algebras.- Boolean Algebras Versus Rings.- The Principle of Duality.- Fields of Sets.- Elementary Relations.- Order.- Infinite Operations.- Topology.- Regular Open Sets.- Subalgebras.- Homomorphisms.- Extensions of Homomorphisms.- Atoms.- Finite Boolean Algebras.- Atomless Boolean Algebras.- Congruences and Quotients.- Ideals and Filters.- Lattices of Ideals.- Maximal Ideals.- Homomorphism and Isomorphism Theorems.- The Representation Theorem.- Canonical Extensions.- Complete Homomorphisms and Complete Ideals.- Completions.- Products of Algebras.- Isomorphisms of Factors.- Free Algebras.- Boolean s-algebras.- The Countable Chain Condition.- Measure Algebras.- Boolean Spaces.- Continuous Functions.- Boolean Algebras and Boolean Spaces.- Duality for Ideals.- Duality for Homomorphisms.- Duality for Subalgebras.- Duality for Completeness.- Boolean s-spaces.- The Representation of s-algebras.- Boolean Measure Spaces.- Incomplete Algebras.- Duality for Products.- Sums of Algebras.- Isomorphisms of Countable Factors.

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Book SynopsisThe book explores how build a mechanical inferences by making use of arithmetic operations on a string of numbers representing statements. In this way logic is reduced to a branch of the combinatory calculus. It covers the field of traditional logic by showing that any kind of inference can be mechanically reduced to three-variables and two-premise inferences. Meriological inferences can also be easily treated in this way. The book covers the following subjects: structural description of space; three-variable inferences through products, sums, subtractions, and divisions; generalization to n variables; relations; and applications.Table of ContentsStructural Description. Product Inferences. Sums. Subtractions. Divisions. Assessment of All the Previous Inferences. Generalized Representation and Structural Relations. Generalized Inferences. Applications. Conclusions. Bibliography. Author Index. Subject Index.

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Book SynopsisAn introduction to natural language semantics that offers an overview of the empirical domain and an explanation of the mathematical concepts that underpin the discipline.This textbook offers a comprehensive introduction to the fundamentals of those approaches to natural language semantics that use the insights of logic. Many other texts on the subject focus on presenting a particular theory of natural language semantics. This text instead offers an overview of the empirical domain (drawn largely from standard descriptive grammars of English) as well as the mathematical tools that are applied to it. Readers are shown where the concepts of logic apply, where they fail to apply, and where they might apply, if suitably adjusted. The presentation of logic is completely self-contained, with concepts of logic used in the book presented in all the necessary detail. This includes propositional logic, first order predicate logic, generalized quantifier theory, and the Lambek an

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Book SynopsisA Publishers Weekly best book of 2015Trade Review"Quirky recipes, personal anecdotes and a large dollop of equations are the key ingredients in this alternative guide to maths and the scientific process. You should find it as easy as cooking a pie." --The Observer, Tech Monthly (UK) "A curious cookbook for the mathematical omnivore." --The Irish Times (Ireland) "Eugenia Cheng's charming new book embeds math in a casing of wry, homespun metaphors: math is like vegan brownies, math is like a subway map, math is like a messy desk. Cheng is at home with math the way you're at home with brownies, maps, and desks, and by the end of How to Bake Pi, you might be, too." --Jordan Ellenberg, Professor of Mathematics, University of Wisconsin-Madison, and author of How Not to Be Wrong "What a charming and original book! The central analogy -- math is like cooking -- turns out to be surprisingly apt and often funny. Light and tasty, yet so, so good for you, How to Bake Pi is a real treat." --Steven Strogatz, Professor of Mathematics, Cornell University and author of The Joy of x "Cheng is exceptional at translating the abstract concepts of mathematics into ordinary language, a strength aided by a writing style that showcases the workings of her curious, sometimes whimsical mind. This combination allows her to demystify how mathematicians think and work, and makes her love for mathematics contagious." --Publishers Weekly, starred review "An original book using recipes to explain sophisticated math concepts to students and even the math-phobic... [Cheng] is a gifted teacher... A sharp, witty book to press on students and even the teachers of math teachers." --Kirkus Reviews "[A] well-written, easy-to-read book." --Library Journal "[T]his book was fun and covered some cool maths, using some nice analogies, and would serve as a good intro for someone getting into category theory." --The Aperiodical (UK) "Eugenia Cheng offers an entertaining introduction to the beauty of mathematics by drawing on insights from the kitchen. She explains why baking a flourless cake is like geometry and offers puzzles to whet the appetites of maths fans." --Times Educational Supplement (UK) "Cheng never quite overeggs her metaphor of the mathematician as chef...and her tone is clear, clever and friendly. Even at her most whimsical she is rigorous and insightful. Potentially confusing ideas are expressed with a matter-of-fact simplicity... How to Bake Pi is a welcome addition to the popular-math shelf, unusual not only because of its quirky premise but also because Cheng is a woman, a lucid and nimble expositor, and unashamedly proud of her domestic obsessions... It would be wonderful if this book attracted a new audience to the field. And there's no better ambassador (or dinner-party host, I'd wager) than Eugenia Cheng." --Alex Bellos, New York Times Book Review "Invoking plenty of examples from cooking and baking, as well as other everyday-life situations such as calculating a taxi fare, searching for love through online dating services and training for a marathon, [Cheng] explains abstract mathematical ideas--including topology and logic--in understandable ways... Her lively, accessible book demonstrates how important and intriguing such a pursuit can be." --Scientific American "[A] funny and engaging new book." --Simon Worrall, National Geographic News "Why go to all the trouble to write a book to help people understand mathematics? Because, as Cheng observes, 'understanding is power, and if you help someone understand something, you're giving them power.' Read How to Bake Pi and you will, indeed, go away feeling empowered." --Marc Merlin, Medium "In her new book, How to Bake Pi, mathematician/baker Eugenia Cheng offers a novel, mathematical approach to cooking... How to Bake Pi is more than a mathematically-minded cookbook. It is just as much a book about mathematical theory and how we learn it. The premise at the heart of the book is that the problem that stops a cookbook from teaching us how to cook is the same problem that makes math classes so bad at actually teaching us to do math." --Ria Misra, io9 "[Cheng] masterfully describes what mathematics is. This includes careful and motivated descriptions of the ideas and methods of abstractions, generalization, logic, and axiomatization... This book is entertaining, insightful, deep and accessible." --Mathematical Reviews "Through an enthusiasm for cooking and zest for life, the author, a math professor, provides a new way to think about a field we thought we knew." --Chemical Engineering Progress "With this delightfully surprising book, Eugenia Cheng reveals the hidden beauty of mathematics with passion and simplicity. After reading How to Bake Pi, you won't look at math (nor porridge!) in the same way ever again." --Roberto Trotta, Astrophysicist, Imperial College London and author of The Edge of the Sky "Math is a lot like cooking. We start with the ingredients we have at hand, try to cook up something tasty, and are sometimes surprised by the results. Does this seem odd? Maybe in school all you got was stale leftovers! Try something better: Eugenia Cheng is not only an excellent mathematician and pastry chef, but a great writer, too." --John Baez, Professor of Math at the University of California, Riverside "From clotted cream to category theory, neither cookery nor math are what you thought they were. But deep down they're remarkably similar. A brilliant gourmet feast of what math is really about." --Ian Stewart, Emeritus Professor of Mathematics at the University of Warwick, and author of Visions of Infinity and Professor Stewart's Incredible Numbers "[O]ften entertaining...frequently illuminating... [How to Bake Pi] offers enough nourishment for the brain to chew on for a long time." --Columbus Dispatch "This is the best book imaginable to introduce someone who doesn't think they are interested in mathematics at all to some of the deep ideas of category theory, especially if they like to bake." --MAA Reviews "Beginning each chapter with a recipe, Cheng converts the making of lasagna, pudding, cookies, and other comestibles into analogies illuminating the mathematical enterprise. Though these culinary analogies teach readers about particular mathematical principles and processes, they ultimately point toward the fundamental character of mathematics as a system of logic, a system presenting daunting difficulties yet offering rare power to make life easier. Despite her zeal for mathematical logic, Cheng recognizes that such logic begins in faith -- irrational faith -- and ultimately requires poetry and art to complement its findings. A singular humanization of the mathematical project." --Booklist, starred review PRAISE FOR HOW TO BAKE PI: "Cheng demystifies math by using recipes to explain mathematical concepts. Her two passions have a good deal in common: Baking and math are centered on similar principles, Cheng notes here, and her clever guide offers tangible examples of abstract ideas." --New York Times Book Review, Paperback Row "Dr. Cheng...has a knack for brushing aside conventions and edicts, like so many pie crumbs from a cutting board. She is a theoretical mathematician who works in a rarefied field called category theory, which is so abstract that 'even some pure mathematicians think it goes too far,' Dr. Cheng said. At the same time, Dr. Cheng is winning fame as a math popularizer, convinced that the pleasures of math can be conveyed to the legions of numbers-averse humanities majors still recovering from high school algebra. She has been featured on shows like Late Night With Stephen Colbert and her online math tutorials have been viewed more than a million times." --Natalie Angier, New York Times "Combined with infectious enthusiasm for cooking and a zest for life, Cheng's perspective on math becomes this singular book: a funny, lively, and clear journey no popular book on math has explored before. How to Bake Pi...will dazzle, amuse, and enlighten." --Gambit Weekly "[Cheng's] book, a very gentle introduction to the main ideas of mathematics in general and category theory in particular, exudes enthusiasm for mathematics, teaching, and creative recipes. Category theory is dangerously abstract, but Cheng's writing is down-to-earth and friendly. She's the kind of person you'd want to talk to at a party, whether about math, food, music, or just the weather... Cheng's cheerful, accessible writing and colorful examples make How to Bake Pi an entertaining introduction to the fundamentals of abstract mathematical thinking." --Evelyn Lamb, Scientific American's Roots of Unity blog "[A] slyly illuminating dispatch on the deep meaning of mathematics... Cheng manages to do for us what the mathematician Keith Devlin has said mathematicians do for themselves: she compels us to see numbers and symbols as vivid characters in an ongoing drama, a narrative in which we are alternately observers and participants." --Natalie Angier, The American Scholar

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Book Synopsis

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Book SynopsisPresents the history of a critical period in mathematics that includes accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. This work provides surveys of many related topics and figures of the late nineteenth century.Trade Review"Grattan-Guiness's uniformly interesting and valuable account of the interwoven development of logic and related fields of mathematics ... between 1870 and 1940 presents a significantly revised analysis of the history of the period... [His] book is important because it supplies what has been lacking: a full account of the period from a primary mathematical perspective."--James W. Van Evra, IsisTable of ContentsCHAPTER 1 Explanations 1.1 Sallies 3 1.2 Scope and limits of the book 3 1.2.1 An outline history 3 1.2.2 Mathematical aspects 4 1.2.3 Historical presentation 6 1.2.4 Other logics, mathematics and philosophies 7 1.3 Citations, terminology and notations 1.3.1 References and the bibliography 9 1.3.2 Translations, quotations and notations 10 1.4 Permissions and acknowledgements 11 CHAPTER 2 Preludes: Algebraic Logic and Mathematical Analysis up to 1870 2.1 Plan of the chapter 14 2.2 'Logique' and algebras in French mathematics 14 2.2.1 The 'logique' and clarity of 'ideologie' 14 2.2.2 Lagrange's algebraic philosophy 15 2.2.3 The many senses of 'analysis' 17 2.2.4 Two Lagrangian algebras: functional equations and differential operators 17 2.2.5 Autonomy for the new algebras 19 2.3 Some English algebraists and logicians 20 2.3.1 A Cambridge revival: the 'Analytical Society, Lacroix, and the professing of algebras 20 2.3.2 The advocacy of algebras by Babbage, Herschel and Peacock 20 2.3.3 An Oxford movement: Whately and the professing of logic 22 2.4 A London pioneer: De Morgan on algebras and logic 25 2.4.1 Summary of his life 25 2.4.2 De Morgan's philosophies of algebra 25 2.4.3 De Morgan's logical career 26 2.4.4 De Morgan's contributions to the foundations of logic 27 2.4.5 Beyond the syllogism 29 2.4.6 Contretemps over 'the quantification of the predicate' 30 2.4.7 The logic of two place relations, 1860 32 2.4.8 Analogies between logic and mathematics 35 2.4.9 De Morgan's theory of collections 36 2.5 A Lincoln outsider: Boole on logic as applied mathematics 37 2.5.1 Summary of his career 37 2.5.2 Boole's 'general method in analysis' 1844 39 2.5.3 The mathematical analysis of logic, 1847. 'elective symbols' and laws 40 2.5.4 'Nothing' and the 'Universe' 42 2.5.5 Propositions, expansion theorems, and solutions 43 2.5.6 The laws of thought, 1854: modified principles and extended methods 46 2.5.7 Boole's new theory of propositions 49 2.5.8 The character of Boole's system 50 2.5.9 Boole's search for mathematical roots 53 2.6 The semi-followers of Boole 54 2.6.1 Some initial reactions to Boole's theory 54 2.6.2 The reformulation by Jevons 56 2.6.3 Jevons versus Boole 59 2.6.4 Followers of Boole and/or Jevons 60 2.7 Cauchy, Weierstrass and the rise of mathematical analysis 63 2.7.1 Different traditions in the calculus 63 2.7.2 Cauchy and the Ecole Polytechnique 64 2.7.3 The gradual adoption and adaptation of Cauchy's new tradition 67 2.7.4 The refinements of Weierstrass and his followers 68 2.8 Judgement and supplement 70 2.8.1 Mathematical analysis versus algebraic logic 70 2.8.2 The places of Kant and Bolzano 71 CHAPTER 3 Cantor: Mathematics as Mengenlehre 3.1 Prefaces 75 3.1.1 Plan of the chapter 75 3.1.2 Cantor's career 75 3.2 The launching of the Mengenlehre, 1870-1883 79 3.2.1 Riemann's thesis: the realm of discontinuous functions 79 3.2.2 Heine on trigonometric series and the real line, 1870-1872 81 3.2.3 Cantor's extension of Heine's findings, 1870-1872 83 3.2.4 Dedekind on irrational numbers, 1872 85 3.2.5 Cantor on line and plane, 1874-1877 88 3.2.6 Infinite numbers and the topology of linear sets, 1878-1883 89 3.2.7 The Grundlagen, 1883: the construction of number-classes 92 3.2.8 The Grundlagen: the definition of continuity 95 3.2.9 The successor to the Grundlagen, 1884 96 3.3 Cantor's Acta mathematica phase, 1883-1885 97 3.3.1 Mittag-Lefler and the French translations, 1883 97 3.3.2 Unpublished and published 'communications' 1884-1885 98 3.3.3 Order-types and partial derivatives in the 'communications' 100 3.3.4 Commentators on Cantor, 1883-1885 102 3.4 The extension of the Mengenlehre, 1886-1897 103 3.4.1 Dedekind's developing set theory, 1888 103 3.4.2 Dedekind's chains of integers 105 3.4.3 Dedekind's philosophy of arithmetic 107 3.4.4 Cantor's philosophy of the infinite, 1886-1888 109 3.4.5 Cantor's new definitions of numbers 110 3.4.6 Cardinal exponentiation: Cantor's diagonal argument, 1891 110 3.4.7 Transfinite cardinal arithmetic and simply ordered sets, 1895 112 3.4.8 Transfinite ordinal arithmetic and well-ordered sets, 1897 114 3.5 Open and hidden questions in Cantor's Mengenlehre 114 3.5.1 Well-ordering and the axioms of choice 114 3.5.2 What was Cantor's 'Cantor's continuum problem'? 116 3.5.3 "Paradoxes" and the absolute infinite 117 3.6 Cantor's philosophy of mathematics 119 3.6.1 A mixed position 119 3.6.2 (No) logic and metamathematics 120 3.6.3 The supposed impossibility of infinitesimals 121 3.6.4 A contrast with Kronecker 122 3.7 Concluding comments: the character of Cantor's achievements 124 CHAPTER 4 Parallel Processes in Set Theory, Logics and Axiomatics, 1870s-1900s 4.1 Plans for the chapter 126 4.2 The splitting and selling of Cantor's Mengenlehre 126 4.2.1 National and international support 126 4.2.2 French initiatives, especially from Borel 127 4.2.3 Couturat outlining the infinite, 1896 129 4.2.4 German initiatives from Mein 130 4.2.5 German proofs of the Schroder-Bernstein theorem 132 4.2.6 Publicity from Hilbert, 1900 134 4.2.7 Integral equations and functional analysis 135 4.2.8 Kempe on 'mathematical form' 137 4.2.9 Kempe-who? 139 4.3 American algebraic logic: Peirce and his followers 140 4.3.1 Peirce, published and unpublished 141 4.3.2 Influences on Peirre's logic: father's algebras 142 4.3.3 Peirce's first phase: Boolean logic and the categories, 1867-1868 144 4.3.4 Peirce's virtuoso theory of relatives, 1870 145 4.3.5 Peirce's second phase, 1880: the propositional calculus 147 4.3.6 Peirre's second phase, 1881: finite and infinite 149 4.3.7 Peirce's students, 1883: duality, and 'Quantifying' a proposition 150 4.3.8 Peirre on 'icons' and the order of 'quantifiers; 1885 153 ~~~ 4.3.9 The Peirceans in the 1890s 154 4.4 German algebraic logic: from the Grassmanns to Schr6der 156 4.4.1 The Grassmanns on duality 156 4.4.2 Schroder's Grassmannian phase 159 4.4.3 Schroder's Peirrean 'lectures' on logic 161 4.4.4 Schrrider's first volume, 1890 161 4.4.5 Part of the second volume, 1891 167 4.4.6 Schroder's third volume, 1895: the 'logic of relatives' 170 4.4.7 Peirce on and against Schroder in The monist, 1896-1897 172 4.4.8 Schroder on Cantorian themes, 1898 174 4.4.9 The reception and publication of Schroder in the 1900s 175 4.5 Frege: arithmetic as logic 177 4.5.1 Frege and Frege' 177 4.5.2 The 'concept-script' calculus of Frege's 'pure thought; 1879 179 4.5.3 Frege's arguments for logicising arithmetic, 1884 183 4.5.4 Keny's conception of Fregean concepts in the mid 1880s 187 4.5.5 Important new distinctions in the early 1890s 187 4.5.6 The 'fundamental laws' of logicised arithmetic, 1893 191 4.5.7 Frege's reactions to others in the later 1890s 194 4.5.8 More 'fundamental laws' of arithmetic, 1903 195 4.5.9 Frege, Korselt and Thomae on the foundations of arithmetic 197 4.6 Husserl: logic as phenomenology 199 4.6.1 A follower of Weierstrass and Cantor 199 4.6.2 The phenomenological 'philosophy of arithmetic; 1891 201 4.6.3 Reviews by Frege and others 203 4.6.4 Husserl's 'logical investigations; 1900-1901 204 4.6.5 Husserl's early talks in Gottingen, 1901 206 4.7 Hilbert: early proof and model theory, 1899-1905 207 4.7.1 Hilbert's growing concern with axiomatics 207 4.7.2 Hilbert's diferent axiom systems for Euclidean geometry, 1899-1902 208 4.7.3 From German completeness to American model theory 209 4.7.4 Frege, Hilbert and Korselt on the foundations of geometries 212 4.7.5 Hilbert's logic and proof theory, 1904-1905 213 4.7.6 Zermelo's logic and set theory, 1904-1909 216 CHAPTER 5 Peano: the Formulary of Mathematics 5.1 Prefaces 219 5.1.1 Plan of the chapter 219 5.1.2 Peano's career 219 5.2 Formalising mathematical analysis 221 5.2.1 Improving Genocchi, 1884 221 5.2.2 Developing Grassmann's 'geometrical calculus; 1888 223 5.2.3 The logistic of arithmetic, 1889 225 5.2.4 The logistic of geometry, 1889 229 5.2.5 The logistic of analysis, 1890 230 5.2.6 Bettazzi on magnitudes, 1890 232 5.3 The Rivista: Peano and his school, 1890-1895 232 5.3.1 The 'society of mathematicians' 232 5.3.2 'Mathematical logic, 1891 234 5.3.3 Developing arithmetic, 1891 235 5.3.4 Infinitesimals and limits, 1892-1895 236 5.3.5 Notations and their range, 1894 237 5.3.6 Peano on definition by equivalence classes 239 5.3.7 Burali-Forti's textbook, 1894 240 5.3.8 Burali-Forti's research, 1896-1897 241 5.4 The Formulaire and the Rivista, 1895-1900 242 5.4.1 The first edition of the Formulaire, 1895 242 5.4.2 Towards the second edition of the Formulaire, 1897 244 5.4.3 Peano on the eliminability of 'the' 246 5.4.4 Frege versus Peano on logic and definitions 247 5.4.5 Schroder's steamships versus Peano's sailing boats 249 5.4.6 New presentations of arithmetic, 1898 251 5.4.7 - Padoa on classhoody 1899 253 5.4.8 Peano's new logical summary, 1900 254 5.5 Peanists in Paris, August 1900 255 5.5.1 An Italian Friday morning 255 5.5.2 Peano on definitions 256 5.5.3 Burali-Forti on definitions of numbers 257 5.5.4 Padoa on definability and independence 259 5.5.5 Pieri on the logic of geometry 261 5.6 Concluding comments: the character of Peano's achievements 262 5.6.1 Peano's little dictionary, 1901 262 5.6.2 Partly grasped opportunities 264 5.6.3 Logic without relations 266 CHAPTER 6 Russell's Way In: From Certainty to Paradoxes, 1895-1903 6.1 Prefaces 268 6.1.1 Plans for two chapters 268 6.1.2 Principal sources 269 6.1.3 Russell as a Cambridge undergraduate, 1891-1894 271 6.1.4 Cambridge philosophy in the 1890s 273 6.2 Three philosophical phases in the foundation of mathematics, 1895-1899 274 6.2.1 Russell's idealist axiomatic geometries 275 6.2.2 The importance of axioms and relations 276 6.2.3 A pair of pas de deux with Paris: Couturat and Poincare on geometries 278 6.2.4 The emergence of "itehead, 1898 280 6.2.5 The impact of G. E. Moore, 1899 282 6.2.6 Three attempted books, 1898-1899 283 6.2.7 Russell's progress with Cantor's Mengenlehre, 1896-1899 285 6.3 From neo-Hegelianism towards 'Principles', 1899-1901 286 6.3.1 Changing relations 286 6.3.2 Space and time, absolutely 288 6.3.3 'Principles of Mathematics, 1899-1900 288 6.4 The first impact of Peano 290 6.4.1 The Paris Congress of Philosophy, August 1900: Schroder versus Peano on 'the' 290 6.4.2 Annotating and popularising in the autumn 291 6.4.3 Dating the origins of Russell's logicism 292 6.4.4 Drafting the logic of relations, October 1900 296 6.4.5 Part 3 of The principles, November 1900: quantity and magnitude 298 6.4.6 Part 4, November 1900: order and ordinals 299 6.4.7 Part 5, November 1900: the transfinite and the continuous 300 6.4.8 Part 6, December 1900: geometries in space 301 6.4.9 Whitehead on 'the algebra of symbolic logic, 1900 302 6.5 Convoluting towards logicism, 1900-1901 303 6.5.1 Logicism as generalised metageometry, January 1901 303 6.5.2 The first paper for Peano, February 1901: relations and numbers 305 6.5.3 Cardinal arithmetic with "itehead and Russell, June 1901 307 6.5.4 The second paper for Peano, March August 1901: set theory with series 308 6.6 From 'fallacy' to 'contradiction', 1900-1901 310 6.6.1 Russell on Cantor's 'fallacy; November 1900 310 6.6.2 Russell's switch to a 'contradiction' 311 6.6.3 Other paradoxes: three too large numbers 312 6.6.4 Three passions and three calamities, 1901-1902 314 6.7 Refining logicism, 1901-1902 315 6.7.1 Attempting Part 1 of The principles, May 1901 315 6.7.2 Part 2, June 1901: cardinals and classes 316 6.7.3 Part 1 again, April-May 1902: the implicational logicism 316 6.7.4 Part 1: discussing the indefinables 318 6.7.5 Part 7, June 1902: dynamics without statics; and within logic? 322 6.7.6 Sort-of finishing the book 323 6.7.7 The first impact of Frege, 1902 323 6.7.8 AppendixA on Frege 326 6.7.9 Appendix B: Russell's first attempt to solve the paradoxes 327 6.8 The roots of pure mathematics? Publishing The principles at last, 1903 328 6.8.1 Appearance and appraisal 328 6.8.2 A gradual collaboration with Whitehead 331 CHAPTER 7 Russell and Whitehead Seek the Principia Mathematica, 1903-1913 7.1 Plan of the chapter 333 7.2 Paradoxes and axioms in set theory, 1903-1906 333 7.2.1 Uniting the paradoxes of sets and numbers 333 7.2.2 New paradoxes, mostly of naming 334 7.2.3 The paradox that got away: heterology 336 7.2.4 Russell as cataloguer of the paradoxes 337 7.2.5 Controversies over axioms of choice, 1904 339 7.2.6 Uncovering Russell's 'multiplicative axiom, 1904 340 7.2.7 Keyser versus Russell over infinite classes, 1903-1905 342 7.3 The perplexities of denoting, 1903-1906 342 7.3.1 First attempts at a general system, 1903-1905 342 7.3.2 Propositional functions, reducible and identical 344 7.3.3 The mathematical importance of definite denoting functions 346 7.3.4 'On denoting' and the complex, 1905 348 7.3.5 Denoting, quantification and the mysteries of existence 350 7.3.6 Russell versus MacColl on the possible, 1904-1908 351 7.4 From mathematical induction to logical substitution, 1905-1907 354 7.4.1 Couturat's Russellian principles 354 7.4.2 A second pas de deux with Paris: Boutroux and Poincare on logicism 355 7.4.3 Poincare on the status of mathematical induction 356 7.4.4 Russell's position paper, 1905 357 7.4.5 Poincare and Russell on the vicious circle principle, 1906 358 7.4.6 The rise of the substitutional theory, 1905-1906 360 7.4.7 The fall of the substitutional theory, 1906-1907 362 7.4.8 Russell's substitutional propositional calculus 364 7.5 Reactions to mathematical logic and logicism, 1904-1907 366 7.5.1 The International Congress of Philosophy, 1904 366 7.5.2 German philosophers and mathematicians, especially Schonflies 368 7.5.3 Activities among the Peanists 370 7.5.4 American philosophers: Royce and Dewey 371 7.5.5 American mathematicians on classes 373 7.5.6 Huntington on logic and orders 375 7.5.7 Judgements fiom Shearman 376 7.6 Whitehead's role and activities, 1905-1907 377 7.6.1 Whitehead's construal of the 'material world' 377 7.6.2 The axioms of geometries 379 7.6.3 Whitehead's lecture course, 1906-1907 379 7.7 The sad compromise: logic in tiers 380 7.7.1 Rehabilitating propositional functions, 1906-1907 380 7.7.2 Two reflective pieces in 1907 382 7.7.3 Russell's outline of 'mathematical logic, 1908 383 7.8 The forming of Principia mathematica 384 7.8.1 Completing and funding Principia mathematica 384 7.8.2 The Organisation of Principia mathematica 386 7.8.3 The propositional calculus, and logicism 388 7.8.4 The predicate calculus, and descriptions 391 7.8.5 Classes and relations, relative to propositional functions 392 7.8.6 The multiplicative axiom: some uses and avoidance 395 7.9 Types and the treatment of mathematics in Principia mathematica 396 7.9.1 7~pes in orders 396 7.9.2 Reducing the edifice 397 7.9.3 Individuals, their nature and number 399 7.9.4 Cardinals and their finite arithmetic 401 7.9.5 The generalised ordinals 403 7.9.6 The ordinals and the alephs 404 7.9.7 The odd small ordinals 406 7.9.8 Series and continuity 406 7.9.9 Quantity with ratios 408 CHAPTER 8 The Influence and Place of Logicism, 1910-1930 8.1 Plans for two chapters 411 8.2 Whitehead's and Russell's transitions from logic to philosophy, 1910-1916 412 8.2.1 The educational concerns of "itehead, 1910-1916 412 8.2.2 Whitehead on the principles of geometry in the 1910s 413 8.2.3 British reviews of Principia mathematica 415 8.2.4 Russell and Peano on logic, 1911-1913 416 8.2.5 Russell's initial problems with epistemology, 1911-1912 417 8.2.6 Russell's first interactions with Wittgenstein, 1911-1913 418 8.2.7 Russell's confrontation with Wiener, 1913 419 8.3 Logicism and epistemology in America and with Russell, 1914-1921 421 8.3.1 Russell on logic and epistemology at Harvard, 1914 421 8.3.2 Two long American reviews 424 8.3.3 Reactions from Royce students: Sheffer and Lewis 424 8.3.4 Reactions to logicism in New York 428 8.3.5 OtherAmerican estimations 429 8.3.6 Russell's 'logical atomism' and psychology, 1917-1921 430 8.3.7 Russell's 'introduction'to logicism, 1918-1919 432 8.4 Revising logic and logicism at Cambridge, 1917-1925 434 8.4.1 New Cambridge authors, 1917-1921 434 8.4.2 Wittgenstein's 'Abhandlung' and Tractatus, 1921-1922 436 8.4.3 The limitations of Wittgenstein's logic 437 8.4.4 Towards extensional logicism: Russell's revision of Principia mathematica, 1923-1924 440 8.4.5 Ramsey's entry into logic and philosophy, 1920-1923 443 8.4.6 Ramsey's recasting of the theory of types, 1926 444 8.4.7 Ramsey on identity and comprehensive extensionality 446 8.5 Logicism and epistemology in Britain and America, 1921-1930 448 8.5.1 Johnson on logic, 1921-1924 448 8.5.2 Other Cambridge authors, 1923-1929 450 8.5.3 American reactions to logicism in mid decade 452 8.5.4 Groping towards metalogic 454 8.5.5 Reactions in and around Columbia 456 8.6 Peripherals: Italy and France 458 8.6.1 The occasional Italian survey 458 8.6.2 New French attitudes in the Revue 459 8.6.3 Commentaries in French, 1918-1930 461 8.7 German-speaking reactions to logicism, 1910-1928 463 8.7.1 (Neo-)Kantians in the 1910s 463 8.7.2 Phenomenologists in the 1910s 467 8.7.3 Frege's positive and then negative thoughts 468 8.7.4 Hilbert's definitive 'metamathematics; 1917-1930 470 8.7.5 Orders of logic and models of set theory: Lowenheim and Skolem, 1915-1923 475 8.7.6 Set theory and Mengenlehre in various forms 476 8.7.7 Intuitionistic set theory and logic: Brouwer and Weyl, 1910-1928 480 8.7.8 (Neo-)Kantians in the 1920s 484 8.7.9 Phenomenologists in the 1920s 487 8.8 The rise of Poland in the 1920s: the Lvnv-Warsaw school 489 8.8.1 From Lv6v to Warsaw: students of Twardowski 489 8.8.2 Logics with Lukasiewicz and Tarski 490 8.8.3 Russell's paradox and Lesniewski's three systems 492 8.8.4 Pole apart: Chwistek's 'semantic' logicism at Cracov 495 8.9 The rise of Austria in the 1920s: the Schlick circle 497 8.9.1 Formation and influence 497 8.9.2 The impact of Russell, especially upon Camap 499 8.9.3 'Logicism ' in Camap's Abriss, 1929 500 8.9.4 Epistemology in Camap's Aufbau, 1928 502 8.9.5 Intuitionism and proof theory: Brouwer and Godel, 1928-1930 504 CHAPTER 9 Postludes: Mathematical Logic and Logicism in the 1930s 9.1 Plan of the chapter 506 9.2 Godel's incompletability theorem and its immediate reception 507 9.2.1 The consolidation of Schlick's 'Vienna' Circle 507 9.2.2 News from G6del: the Konigsberg lectures, September 1930 508 9.2.3 G6del's incompletability theorem, 1931 509 9.2.4 Effects and reviews of G6del's theorem 511 9.2.5 Zermelo against Godeb the Bad Elster lectures, September 1931 512 9.3 Logic(ism) and epistemology in and around Vienna 513 9.3.1 Carnap for 'metalogic' and against metaphysics 513 9.3.2 Carnap's transformed metalogic: the 'logical syntax of language; 1934 515 9.3.3 Carnap on incompleteness and truth in mathematical theories, 1934-1935 517 9.3.4 Dubislav on definitions and the competing philosophies of mathematics 519 9.3.5 Behmann's new diagnosis of the paradoxes 520 9.3.6 Kaufmann and Waismann on the philosophy of mathematics 521 9.4 Logic(ism) in the U.S.A. 523 9.4.1 Mainly Eaton and Lewis 523 9.4.2 Mainly Weiss and Langer 525 9.4.3 Whitehead's new attempt to ground logicism, 1934 527 9.4.4 The debut of Quine 529 9.4.5 Two journals and an encyclopaedia, 1934-1938 531 9.4.6 Carnap's acceptance of the autonomy of semantics 533 9.5 The battle of Britain 535 9.5.1 The campaign of Stebbing for Russell and Carnap 535 9.5.2 Commentary from Black and Ayer 538 9.5.3 Mathematicians-and biologists 539 9.5.4 Retiring into philosophy: Russell's return, 1936-1937 542 9.6 European, mostly northern 543 9.6.1 Dingler and Burkamp again 543 9.6.2 German proof theory after Godel 544 9.6.3 Scholz's little circle at Munster 546 9.6.4 Historical studies, especially by Jorgensen 547 9.6.5 History philosophy, especially Cavailles 548 9.6.6 Other Francophone figures, especially Herbrand 549 9.6.7 Polish logicians, especially Tarski 551 9.6.8 Southern Europe and its former colonies 553 CHAPTER 10 The Fate of the Search 10.1 Influences on Russell, negative and positive 556 10.1.1 Symbolic logics: living together and living apart 556 10.1.2 The timing and origins of Russell's logicism 557 10.1.3 (Why) was Frege (so) little read in his lifetime? 558 10.2 The content and impact of logicism 559 10.2.1 Russell's obsession with reductionist logic and epistemology 560 10.2.2 The logic and its metalogic 562 10.2.3 The fate of logicism 563 10.2.4 Educational aspects, especially Piaget 566 10.2.5 The role of the U.S.A.: judgements in the Schi1pp series 567 10.3 The panoply of foundations 569 10.4 Sallies 573 CHAPTER 11 Transcription of Manuscripts 11.1 Couturat to Russell, 18 December 1904 574 11.2 Veblen to Russell, 13 May 1906 577 11.3 Russell to Hawtrey, 22 January 1907 (or 1909?) 579 11.4 Jourdain's notes on Wittgenstein's first views on Russell's paradox, April 1909 580 11.5 The application of Whitehead and Russell to the Royal Society, late 1909 581 11.6 Whitehead to Russell, 19 January 1911 584 11.7 Oliver Strachey to Russell, 4 January 1912 585 11.8 Quine and Russell, June-July 1935 586 11.8.1 Russell to Quine, 6 June 1935 587 11.8.2 Quine to Russell, 4 July 1935 588 11.9 Russell to Henkin, 1 April 1963 592 BIBLIOGRAPHY 594 INDEX 671

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Book SynopsisProvides a unique and methodologically consistent treatment of various areas of fuzzy modeling and includes the results of mathematical fuzzy logic and linguistics This book is the result of almost thirty years of research on fuzzy modeling. It provides a unique view of both the theory and various types of applications.Table of ContentsPreface xiii Acknowledgments xv About the Companion Website xvii Part I Fundamentals of Fuzzy Modeling 1 1 What is Fuzzy Modeling 3 2 Overview of Basic Notions 11 3 Fuzzy IF-THEN Rules in Approximation of Functions 49 4 Fuzzy Transform 81 5 Fuzzy Natural Logic and Approximate Reasoning 97 6 Fuzzy Cluster Analysis 137 Part II Selected Applications 149 7 Fuzzy/Linguistic Control and Decision-Making 151 8 F-Transform in Image Processing 189 9 Analysis and Forecasting of Time Series 209

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Book SynopsisCrossing the River with Dogs: Problem Solving for College Students, 3rd Edition promotes the philosophy that students learn best by working in groups and the skills required for real workplace problem solving are those skills of collaboration. The text aims to improve students' writing, oral communication, and collaboration skills while teaching mathematical problem-solving strategies. Focusing entirely on problem solving and using issues relevant to college students for examples, the authors continue their approach of explaining classic as well as non-traditional strategies through dialogs among fictitious students. This text is appropriate for a problem solving, quantitative reasoning, liberal arts mathematics, mathematics for elementary teachers, or developmental mathematics course.Table of ContentsPreface vii Instructor Resources x Acknowledgments xi Introduction 1 1 Draw a Diagram 11 2 Make a Systematic List 27 3 Eliminate Possibilities 47 4 Use Matrix Logic 73 5 Look for a Pattern 115 6 Guess and Check 145 7 Identify Subproblems 175 8 Analyze the Units 199 9 Solve an Easier Related Problem 233 10 Create a Physical Representation 267 11 Work Backwards 297 12 Draw Venn Diagrams 323 13 Convert to Algebra 351 14 Evaluate Finite Differences 383 15 Organize Information in More Ways 417 16 Change Focus in More Ways 447 17 Visualize Spatial Relationships 473 Appendix 503 Unit Analysis Adding, Subtracting, Multiplying, and Dividing Fractions Area and Volume Formulas Properties of Triangles Properties of Numbers Glossary 511 Bibliography 519 Index of Problem Titles 521 General Index 529 Photo Credits 539 Answers to More Practice Problems 541

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Book SynopsisA Petri net is a mathematical representation of a network. This book first introduces the basic models including time and stochastic extensions, in particular place-transition and high level Petri nets. Their modeling and design capabilities are illustrated by a set of representations of interest in operating and communication systems. The volume then addresses the related verification problems and proposes corresponding solutions by introducing the main notions needed to fully understand the behavior and properties behind Petri nets. Particular attention is devoted to how systems can be fully represented and analyzed in terms of their behavioral, time, and stochastic aspects by using the same formal approach and semantic basis. Finally, illustrative examples are presented in the important fields of interoperability in telecommunication services, programming languages, multimedia architectures, manufacturing systems, and communication protocols.Trade Review"We think that this volume should greatly help any designer to build the new forthcoming generation of distributed systems." (Mathematical Reviews, 2011) Table of ContentsPreface xv Introduction xvii PART 1. FUNDAMENTAL MODELS 1 Chapter 1. Basic Semantics 3 Michel DIAZ Chapter 2. Application of Petri Nets to Communication Protocols 27 Michel DIAZ Chapter 3. Analysis Methods for Petri 41 Serge HADDAD and François VERNADAT Chapter 4. Decidability and Complexity of Petri Net Problems 87 Serge HADDAD Chapter 5. Time Petri Nets 123 Bernard BERTHOMIEU, Marc BOYER and Michel DIAZ Chapter 6. Temporal Composition and Time Stream Petri Nets 163 Michel DIAZ and Patrick SÉNAC Chapter 7. High Level Petri Nets 185 Claude GIRAULT and Jean-François PRADAT-PEYRE Chapter 8. Analysis of High Level Petri Nets 221 Claude GIRAULT and Jean-François PRADAT-PEYRE Chapter 9. Stochastic Petri Nets 269 Serge HADDAD and Patrice MOREAUX Chapter 10. Stochastic Well-formed Petri Nets 303 Serge HADDAD and Patrice MOREAUX Chapter 11. Tensor Methods and Stochastic Petri Nets 321 Serge HADDAD and Patrice MOREAUX PART 2. VERIFICATION AND APPLICATION OF PETRI NETS 347 Chapter 12. Verification of Specific Properties 349 Serge HADDAD and François VERNADAT Chapter 13. Petri Net Unfoldings – Properties 415 Jean-Michel COUVREUR and Denis POITRENAUD Chapter 14. Symmetry and Temporal Logic 435 Serge HADDAD and Jean-Michel ILIÉ Chapter 15. Hierarchical Time Stream Petri Nets 461 Patrick SÉNAC and Michel DIAZ Chapter 16. Petri Nets and Linear Logic 481 Brigitte PRADIN, Robert VALETTE and Nicolas RIVIÈRE Chapter 17. Modeling of Multimedia Architectures: the Case of Videoconferencing with Guaranteed Quality of Service 501 Philippe OWEZARSKI and Marc BOYER Chapter 18 Performance Evaluation in Manufacturing Systems 527 Isabel DEMONGODIN, Nathalie SAUER and Laurent TRUFFET Conclusion 577 List of Authors 579 Index 581

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Book SynopsisThis monograph provides an overview of developments in group theory motivated by model theory by key international researchers in the field. Topics covered include: stable groups and generalizations, model theory of nonabelian free groups and of rigid solvable groups, pseudofinite groups, approximate groups, topological dynamics, groups interpreting the arithmetic. The book is intended for mathematicians and graduate students in group theory and model theory. The book follows the course of the GAGTA (Geometric and Asymptotic Group Theory with Applications) conference series. The first book, "Complexity and Randomness in Group Theory. GAGTA book 1," can be found here: http://www.degruyter.com/books/978-3-11-066491-1 .

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Book SynopsisPresents an area of mathematical research that combines topology, geometry, and logic. This book seeks to explain and illustrate the implications of the general principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity.Trade Review"This is a terrific book. It does no less than introduce an entire new field of mathematics - a truly astounding development. It will be widely read, I think, as much because of the masterful exposition as for the beautiful mathematics. Weinberger gives very clear and accessible descriptions of all the relevant tools from computability, topology, and geometry, in a friendly and engaging style. He has done the mathematical community a great service indeed." - Robin Forman, Rice University; "This book represents a very exciting new area of research at the interface of topology and logic. Written in a quite readable style, and presenting the more accessible cases in detail while giving references for the more involved results, it is a book whose methods and ideas will surely have many more significant applications over the next several years." - Kevin M. Whtye, University of Illinois at Chicago"

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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 SynopsisPresents various theorems of the theory of sets along with complete proofs. This book discusses strengthening of theorems, the simplification of proofs, and the removal of unnecessary hypotheses.Trade ReviewAn indispensible book for all those interested in the theory of sets and the allied branches of real variable theory."" — Bulletin of the AMSTable of Contents Sets and the Combining of Sets: 1.1 Sets 1.2 Functions 1.3 Sum and intersection 1.4 Product and power Cardinal Numbers: 2.5 Comparison of sets 2.6 Sum, product, and power 2.7 The scale of cardinal numbers 2.8 The elementary cardinal numbers Order Types: 3.9 Order 3.10 Sum and product 3.11 The types $\aleph_0$ and $\aleph$ Ordinal Numbers: 4.12 The well-ordering theorem 4.13 The comparability of ordinal numbers 4.14 The combining of ordinal numbers 4.15 The alefs 4.16 The general concept of product Systems of Sets: 5.17 Rings and fields 5.18 Borel systems 5.19 Suslin sets Point Sets: 6.20 Distance 6.21 Convergence 6.22 Interior points and border points 6.23 The $\alpha, \beta$, and $\gamma$ points 6.24 Relative and absolute concepts 6.25 Separable spaces 6.26 Complete spaces 6.27 Sets of the first and second categories 6.28 Spaces of sets 6.29 Connectedness Point Sets and Ordinal Numbers: 7.30 Hulls and kernels 7.31 Further applications of ordinal numbers 7.32 Borel and Suslin sets 7.33 Existence proofs 7.34 Criteria for Borel sets Mappings of Two Spaces: 8.35 Continuous mappings 8.36 Interval-images 8.37 Images of Suslin sets 8.38 Homeomorphism 8.39 Simple curves 8.40 Topological spaces Real Functions: 9.41 Functions and inverse image sets 9.42 Functions of the first class 9.43 Baire functions 9.44 Sets of convergence Supplement: 10.45 The Baire condition 10.46 Half-schlicht mappings Appendixes Bibliography Further references Index

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Book SynopsisHilbert's tenth problem is one of 23 problems proposed by David Hilbert in 1900. It asked for a general algorithm to determine if a given Diophantine equation has a solution in integers. It was finally resolved by Julia Robinson, Martin Davis, Hilary Putnam, and finally Yuri Matiyasevich in 1970. This book is an exposition of their achievement.Table of Contents Introduction Cantor and infinity Axiomatic set theory Elementary number theory Computability and provability Hilbert's tenth problem Applications of Hilbert's tenth problem Hilbert's tenth problem over number fields Background material Bibliography Index

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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 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 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 SynopsisThis monograph covers the recent major advances in various areas of set theory. From the reviews: "One of the classical textbooks and reference books in set theory....The present ‘Third Millennium’ edition...is a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master....This well-written book promises to influence the next generation of set theorists, much as its predecessor has done." --MATHEMATICAL REVIEWSTrade ReviewFrom the reviews of the third edition: "Thomas Jech’s text has long been considered a classic survey of the state of the set theory … . As every logician will know, this is a work of extraordinary scholarship, essential for any graduate logician who needs to know where the current boundaries of research are situated. Each chapter ends with a valuable historical survey and there is an extensive bibliography. This will continue to be the bible for set theorists in the new century." (Gerry Leversha, The Mathematical Gazette, March, 2005) "The book does masterly what it is supposed to do. … every mathematician who wishes to refresh his knowledge of set theory will read it with pleasure. … They will also find historical notes, and precise references … . A very comprehensive bibliography, and detailed indexes complete the work. This book fills a serious gap in the literature and there is no doubt that it will become a standard reference … . One can strongly recommend its acquisition for any mathematical library." (Jean-Roger Roisin, Bulletin of the Belgian Mathematical Society, Vol. 11 (3), 2004) "One of the classical textbooks and reference books in set theory is Jech’s Set Theory. … The present ‘Third Millennium’ edition … is a whole new book. In three parts the author offers us what in his view every young set theorist should learn and master. … This well-written book promises to influence the next generation of set theorists, much as its predecessor has done over the last quarter of a century." (Eva Coplakova, Mathematical Reviews, 2004 g) "Jech’s book, ‘Set Theory’ has been a standard reference for over 25 years. This ‘Third Millennium Edition’, not only includes all the materials in the first two editions, but also covers recent developments of set theory during the last 25 years. We believe that this new version will become a standard reference on set theory for the next few years." (Guohua Wu, New Zealand Mathematical Society Newsletter, April, 2004) "Jech’s classic monograph has been a standard reference for a generation of set theorists. Though … labeled ‘The Third Millennium Edition’, the present work is in fact a new book. ... Even sections presenting older results have been rewritten and modernized. Exercises have been moved to the end of each section. The bibliography, the section on notation, and the index have been considerably expanded as well. This new edition will certainly become a standard reference on set theory for years to come." (Jörg D. Brendle, Zentralblatt MATH, Vol. 1007, 2003) "Thomas Jech’s Set Theory contains the most comprehensive treatment of the subject in any one volume. The present third edition is a revised and expanded version … . The third edition has three parts. The first, Jech says, every student of set theory should learn, the second every set theorist should master and the third consists of various results reflecting ‘the state of the art of set theory at the turn of the new millennium’. This last part especially contains a lot of new material." (Martin Bunder, The Australian Mathematical Society Gazette, Vol. 30 (2), 2003)Table of ContentsBasic Set Theory.- Axioms of Set Theory.- Ordinal Numbers.- Cardinal Numbers.- Real Numbers.- The Axiom of Choice and Cardinal Arithmetic.- The Axiom of Regularity.- Filters, Ultrafilters and Boolean Algebras.- Stationary Sets.- Combinatorial Set Theory.- Measurable Cardinals.- Borel and Analytic Sets.- Models of Set Theory.- Advanced Set Theory.- Constructible Sets.- Forcing.- Applications of Forcing.- Iterated Forcing and Martin’s Axiom.- Large Cardinals.- Large Cardinals and L.- Iterated Ultrapowers and L[U].- Very Large Cardinals.- Large Cardinals and Forcing.- Saturated Ideals.- The Nonstationary Ideal.- The Singular Cardinal Problem.- Descriptive Set Theory.- The Real Line.- Selected Topics.- Combinatorial Principles in L.- More Applications of Forcing.- More Combinatorial Set Theory.- Complete Boolean Algebras.- Proper Forcing.- More Descriptive Set Theory.- Determinacy.- Supercompact Cardinals and the Real Line.- Inner Models for Large Cardinals.- Forcing and Large Cardinals.- Martin’s Maximum.- More on Stationary Sets.

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