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This introductory treatment covers the basic concepts and machinery of stability theory. Lemmas, corollaries, proofs, and notes assist readers in working through and understanding the material and applications. Full of examples, theorems, propositions, and problems, it is suitable for graduate students in logic and mathematics, professional mathematicians, and computer scientists. Chapter 1 introduces the notions of definable type, heir, and coheir....
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Coherent, well-organized text familiarizes readers with complete theory of logical inference and its applications to math and the empirical sciences. Part I deals with formal principles of inference and definition. Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Last section introduces numerous examples of axiomatically formulated theories.
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Designed as a methodDesigned as a method for teaching correct mathematical thinking to high school students, this book contains a brilliantly constructed series of what the authors call "lapses," erroneous statements that are part of a larger mathematical argument. These lapses lead to sophism or mathematical absurdities. The ingenious idea behind this technique is to lead the student deliberately toward a clearly false conclusion. The teacher and...
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This highly technical introduction to formal languages in computer science covers all areas of mainstream formal language theory, including such topics as operations on languages, context-sensitive languages, automata, decidability, syntax analysis, derivation languages, and more. Geared toward advanced undergraduates and graduate students, the treatment examines mathematical topics related to mathematical logic, set theory, and linguistics. All subjects...
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Comprehensive in its selection of topics and results, this self-contained text examines the relative strengths and consequences of the axiom of choice. Each chapter contains several problems, graded according to difficulty, and concludes with some historical remarks. An introduction to the use of the axiom of choice is followed by explorations of consistency, permutation models, and independence. Subsequent chapters examine embedding theorems, models...
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Mathematical induction - along with its equivalents, complete induction and well-ordering, and its immediate consequence, the pigeonhole principle - constitute essential proof techniques. Every mathematician is familiar with mathematical induction, and every student of mathematics requires a grasp of its concepts. This volume provides an introduction and a thorough exposure to these proof techniques. Geared toward students of mathematics at all levels,...
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Category theory has provided the foundations for many of the twentieth century's greatest advances in pure mathematics. This concise, original text for a one-semester course on the subject is derived from courses that author Emily Riehl taught at Harvard and Johns Hopkins Universities. The treatment introduces the essential concepts of category theory: categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions,...
8) Logic
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The second, corrected edition of the first and only complete English translation of Kant's highly influential introduction to philosophy, presenting both the terminological and structural basis for his philosophical system, and offering an invaluable key to his main works, particularly the three Critiques. Extensive editiorial apparatus.
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Hailed by the Bulletin of the American Mathematical Society as "easy to use and a pleasure to read," this research monograph is recommended for students and professionals interested in model theory and definability theory. The sole prerequisite is a familiarity with the basics of logic, model theory, and set theory. The author, Professor of Mathematics at UCLA and Emeritus Professor of Mathematics,University of Athens, Greece, begins with a focus...
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In modern mathematics, both the theory of proof and the derivation of theorems from axioms bear an unquestioned importance. The necessary skills behind these methods, however, are frequently underdeveloped. This book counters that neglect with a rigorous introduction that is simple enough in presentation and context to permit relatively easy comprehension. It comprises the sentential theory of inference, inference with universal quantifiers, and applications...
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Combining stories of great philosophers, quotations, and riddles with the fundamentals of mathematical logic, this new textbook for first courses in mathematical logic was written by the subject's creative master. Raymond Smullyan offers clear, incremental presentations of difficult logic concepts with creative explanations and unique problems related to proofs, propositional logic and first-order logic, undecidability, recursion theory, and other...
12) Topos Theory
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One of the best books on a relatively new branch of mathematics, this volume focuses on how topos theory integrates geometric and logical ideas into the foundations of mathematics and theoretical computer science. Topics include internal category theory, topologies and sheaves, geometric morphisms, natural number objects, cohomology, set theory, and more. 1977 edition.
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Over the past two centuries the field of logic has developed at an explosive pace into new areas far removed from the traditional syllogism and formal proof. The purpose of this well-known introductory treatment is to chart, clearly and lucidly, this new domain of today's vastly sophisticated logic. Author Morris R. Cohen explores "the periphery of logic, the relations of logic to the rest of the universe, the philosophical presuppositions which give...
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This introduction to the main ideas and results of mathematical logic is a serious treatment geared toward non-logicians. Starting with a historical survey of logic in ancient times, it traces the 17th-century development of calculus and discusses modern theories, including set theory, the continuum hypothesis, and other ideas. 1972 edition.
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A classic introduction to mathematical logic from the perspective of category theory, this text is suitable for advanced undergraduates and graduate students and accessible to both philosophically and mathematically oriented readers. Its approach moves always from the particular to the general, following through the steps of the abstraction process until the abstract concept emerges naturally. Beginning with a survey of set theory and its role in...
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In 1931, a young Austrian mathematician published an epoch-making paper containing one of the most revolutionary ideas in logic since Aristotle. Kurt Giidel maintained, and offered detailed proof, that in any arithmetic system, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. The repercussions...
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This third edition of a popular, well-received text offers undergraduates an opportunity to obtain an overview of the historical roots and the evolution of several areas of mathematics. The selection of topics conveys not only their role in this historical development of mathematics but also their value as bases for understanding the changing nature of mathematics. Among the topics covered in this wide-ranging text are: mathematics before Euclid,...
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From one of the founders of symbolic logic comes this collection of writings on logical subjects and related questions of probability. George Boole invented Boolean logic, the basis of modern digital computer logic, for which he is regarded as a founder of the field of computer science. This authoritative compilation of his papers features his most mature thinking on Boolean logic and includes previously unpublished material. Appropriate for upper-level...
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The primary purpose of this undergraduate text is to teach students to do mathematical proofs. It enables readers to recognize the elements that constitute an acceptable proof, and it develops their ability to do proofs of routine problems as well as those requiring creative insights. The self-contained treatment features many exercises, problems, and selected answers, including worked-out solutions. Starting with sets and rules of inference, this...
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Classic introduction to objectives and methods of schools of empiricism and linguistic analysis, especially of the logical positivism derived from the Vienna Circle. Topics: elimination of metaphysics, function of philosophy, nature of philosophical analysis, the a priori, truth and probability, critique of ethics and theology, self and the common world, more.
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