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The Structure Of Models Of Peano Arithmetic
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Book Synopsis Models of Peano Arithmetic by : Richard Kaye
Download or read book Models of Peano Arithmetic written by Richard Kaye and published by . This book was released on 1991 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Non-standard models of arithmetic are of interest to mathematicians through the presence of infinite integers and the various properties they inherit from the finite integers. Since their introduction in the 1930s, they have come to play an important role in model theory, and in combinatorics through independence results such as the Paris-Harrington theorem. This book is an introduction to these developments, and stresses the interplay between the first-order theory, recursion-theoretic aspects, and the structural properties of these models. Prerequisites for an understanding of the text have been kept to a minimum, these being a basic grounding in elementary model theory and a familiarity with the notions of recursive, primitive recursive, and r.e. sets. Consequently, the book is suitable for postgraduate students coming to the subject for the first time, and a number of exercises of varying degrees of difficulty will help to further the reader's understanding.
Book Synopsis The Structure of Models of Peano Arithmetic by : Roman Kossak
Download or read book The Structure of Models of Peano Arithmetic written by Roman Kossak and published by Oxford University Press. This book was released on 2006-06-29 with total page 326 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aimed at graduate students, research logicians and mathematicians, this much-awaited text covers over 40 years of work on relative classification theory for nonstandard models of arithmetic. The book covers basic isomorphism invariants: families of type realized in a model, lattices of elementary substructures and automorphism groups.
Book Synopsis Uncountably Categorical Theories by : Boris Zilber
Download or read book Uncountably Categorical Theories written by Boris Zilber and published by American Mathematical Soc.. This book was released on with total page 132 pages. Available in PDF, EPUB and Kindle. Book excerpt: The 1970s saw the appearance and development in categoricity theory of a tendency to focus on the study and description of uncountably categorical theories in various special classes defined by natural algebraic or syntactic conditions. There have thus been studies of uncountably categorical theories of groups and rings, theories of a one-place function, universal theories of semigroups, quasivarieties categorical in infinite powers, and Horn theories. In Uncountably Categorical Theories , this research area is referred to as the special classification theory of categoricity. Zilber's goal is to develop a structural theory of categoricity, using methods and results of the special classification theory, and to construct on this basis a foundation for a general classification theory of categoricity, that is, a theory aimed at describing large classes of uncountably categorical structures not restricted by any syntactic or algebraic conditions.
Book Synopsis Nonstandard Models of Arithmetic and Set Theory by : Ali Enayat
Download or read book Nonstandard Models of Arithmetic and Set Theory written by Ali Enayat and published by American Mathematical Soc.. This book was released on 2004 with total page 184 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the proceedings of the AMS special session on nonstandard models of arithmetic and set theory held at the Joint Mathematics Meetings in Baltimore (MD). The volume opens with an essay from Haim Gaifman that probes the concept of non-standardness in mathematics and provides a fascinating mix of historical and philosophical insights into the nature of nonstandard mathematical structures. In particular, Gaifman compares and contrasts the discovery of nonstandard models with other key mathematical innovations, such as the introduction of various number systems, the modern concept of function, and non-Euclidean geometries. Other articles in the book present results related to nonstandard models in arithmetic and set theory, including a survey of known results on the Turing upper bounds of arithmetic sets and functions. The volume is suitable for graduate students and research mathematicians interested in logic, especially model theory.
Book Synopsis Predicative Arithmetic. (MN-32) by : Edward Nelson
Download or read book Predicative Arithmetic. (MN-32) written by Edward Nelson and published by Princeton University Press. This book was released on 2014-07-14 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed. Originally published in 1986. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Download or read book Model Theory written by María Manzano and published by Oxford University Press. This book was released on 1999 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: Model theory, which is concerned with the relationship between mathematical structures and logic, now has a wide range of applications in areas such as computing, philosophy, and linguistics. This book, suitable for both mathematicians and students from outside the field, provides a clear and readable introduction to the subject.
Book Synopsis A Shorter Model Theory by : Wilfrid Hodges
Download or read book A Shorter Model Theory written by Wilfrid Hodges and published by Cambridge University Press. This book was released on 1997-04-10 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an up-to-date textbook of model theory taking the reader from first definitions to Morley's theorem and the elementary parts of stability theory. Besides standard results such as the compactness and omitting types theorems, it also describes various links with algebra, including the Skolem-Tarski method of quantifier elimination, model completeness, automorphism groups and omega-categoricity, ultraproducts, O-minimality and structures of finite Morley rank. The material on back-and-forth equivalences, interpretations and zero-one laws can serve as an introduction to applications of model theory in computer science. Each chapter finishes with a brief commentary on the literature and suggestions for further reading. This book will benefit graduate students with an interest in model theory.
Book Synopsis Principia Mathematica by : Alfred North Whitehead
Download or read book Principia Mathematica written by Alfred North Whitehead and published by . This book was released on 1910 with total page 688 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Model Theory : An Introduction by : David Marker
Download or read book Model Theory : An Introduction written by David Marker and published by Springer Science & Business Media. This book was released on 2006-04-06 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: Assumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structures
Book Synopsis The Structure of Models of Peano Arithmetic by : Roman Kossak
Download or read book The Structure of Models of Peano Arithmetic written by Roman Kossak and published by Clarendon Press. This book was released on 2006-06-29 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: Aimed at graduate students and research logicians and mathematicians, this much-awaited text covers over forty years of work on relative classification theory for non-standard models of arithmetic. With graded exercises at the end of each chapter, the book covers basic isomorphism invariants: families of types realized in a model, lattices of elementary substructures and automorphism groups. Many results involve applications of the powerful technique of minimal types due to Haim Gaifman, and some of the results are classical but have never been published in a book form before.
Book Synopsis Metamathematics of First-Order Arithmetic by : Petr Hájek
Download or read book Metamathematics of First-Order Arithmetic written by Petr Hájek and published by Cambridge University Press. This book was released on 2017-03-02 with total page 475 pages. Available in PDF, EPUB and Kindle. Book excerpt: A much-needed monograph on the metamathematics of first-order arithmetic, paying particular attention to fragments of Peano arithmetic.
Book Synopsis The Structure of Models of Peano Arithmetic by : Roman Kossak
Download or read book The Structure of Models of Peano Arithmetic written by Roman Kossak and published by . This book was released on 2006 with total page 311 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Logic of Mathematics by : Zofia Adamowicz
Download or read book Logic of Mathematics written by Zofia Adamowicz and published by John Wiley & Sons. This book was released on 2011-09-26 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: A thorough, accessible, and rigorous presentation of the central theorems of mathematical logic . . . ideal for advanced students of mathematics, computer science, and logic Logic of Mathematics combines a full-scale introductory course in mathematical logic and model theory with a range of specially selected, more advanced theorems. Using a strict mathematical approach, this is the only book available that contains complete and precise proofs of all of these important theorems: * Gödel's theorems of completeness and incompleteness * The independence of Goodstein's theorem from Peano arithmetic * Tarski's theorem on real closed fields * Matiyasevich's theorem on diophantine formulas Logic of Mathematics also features: * Full coverage of model theoretical topics such as definability, compactness, ultraproducts, realization, and omission of types * Clear, concise explanations of all key concepts, from Boolean algebras to Skolem-Löwenheim constructions and other topics * Carefully chosen exercises for each chapter, plus helpful solution hints At last, here is a refreshingly clear, concise, and mathematically rigorous presentation of the basic concepts of mathematical logic-requiring only a standard familiarity with abstract algebra. Employing a strict mathematical approach that emphasizes relational structures over logical language, this carefully organized text is divided into two parts, which explain the essentials of the subject in specific and straightforward terms. Part I contains a thorough introduction to mathematical logic and model theory-including a full discussion of terms, formulas, and other fundamentals, plus detailed coverage of relational structures and Boolean algebras, Gödel's completeness theorem, models of Peano arithmetic, and much more. Part II focuses on a number of advanced theorems that are central to the field, such as Gödel's first and second theorems of incompleteness, the independence proof of Goodstein's theorem from Peano arithmetic, Tarski's theorem on real closed fields, and others. No other text contains complete and precise proofs of all of these theorems. With a solid and comprehensive program of exercises and selected solution hints, Logic of Mathematics is ideal for classroom use-the perfect textbook for advanced students of mathematics, computer science, and logic.
Book Synopsis Computable Structure Theory by : Antonio Montalbán
Download or read book Computable Structure Theory written by Antonio Montalbán and published by Cambridge University Press. This book was released on 2021-06-24 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: In mathematics, we know there are some concepts - objects, constructions, structures, proofs - that are more complex and difficult to describe than others. Computable structure theory quantifies and studies the complexity of mathematical structures, structures such as graphs, groups, and orderings. Written by a contemporary expert in the subject, this is the first full monograph on computable structure theory in 20 years. Aimed at graduate students and researchers in mathematical logic, it brings new results of the author together with many older results that were previously scattered across the literature and presents them all in a coherent framework, making it easier for the reader to learn the main results and techniques in the area for application in their own research. This volume focuses on countable structures whose complexity can be measured within arithmetic; a forthcoming second volume will study structures beyond arithmetic.
Book Synopsis Lectures on the Curry-Howard Isomorphism by : Morten Heine Sørensen
Download or read book Lectures on the Curry-Howard Isomorphism written by Morten Heine Sørensen and published by Elsevier. This book was released on 2006-07-04 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance,minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc.The isomorphism has many aspects, even at the syntactic level:formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc.But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transformsproofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq).This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic.Key features- The Curry-Howard Isomorphism treated as common theme- Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics- Thorough study of the connection between calculi and logics- Elaborate study of classical logics and control operators- Account of dialogue games for classical and intuitionistic logic- Theoretical foundations of computer-assisted reasoning· The Curry-Howard Isomorphism treated as the common theme.· Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics.· Elaborate study of classical logics and control operators.· Account of dialogue games for classical and intuitionistic logic.· Theoretical foundations of computer-assisted reasoning
Book Synopsis Subsystems of Second Order Arithmetic by : Stephen George Simpson
Download or read book Subsystems of Second Order Arithmetic written by Stephen George Simpson and published by Cambridge University Press. This book was released on 2009-05-29 with total page 461 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume examines appropriate axioms for mathematics to prove particular theorems in core areas.
Download or read book Analysis I written by Terence Tao and published by Springer. This book was released on 2016-08-29 with total page 366 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is part one of a two-volume book on real analysis and is intended for senior undergraduate students of mathematics who have already been exposed to calculus. The emphasis is on rigour and foundations of analysis. Beginning with the construction of the number systems and set theory, the book discusses the basics of analysis (limits, series, continuity, differentiation, Riemann integration), through to power series, several variable calculus and Fourier analysis, and then finally the Lebesgue integral. These are almost entirely set in the concrete setting of the real line and Euclidean spaces, although there is some material on abstract metric and topological spaces. The book also has appendices on mathematical logic and the decimal system. The entire text (omitting some less central topics) can be taught in two quarters of 25–30 lectures each. The course material is deeply intertwined with the exercises, as it is intended that the student actively learn the material (and practice thinking and writing rigorously) by proving several of the key results in the theory.
