Set Theoretical Logic The Algebra Of Models
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Set Theoretical Logic The Algebra Of Models
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Author : W Felscher
language : en
Publisher: CRC Press
Release Date : 2000-05-30
Set Theoretical Logic The Algebra Of Models written by W Felscher and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-05-30 with Mathematics categories.
This is an introduction to mathematical logic in which all the usual topics are presented: compactness and axiomatizability of semantical consequence, Löwenheim-Skolem-Tarski theorems, prenex and other normal forms, and characterizations of elementary classes with the help of ultraproducts. Logic is based exclusively on semantics: truth and satisfiability of formulas in structures are the basic notions. The methods are algebraic in the sense that notions such as homomorphisms and congruence relations are applied throughout in order to gain new insights. These concepts are developed and can be viewed as a first course on universal algebra. The approach to algorithms generating semantical consequences is algebraic as well: for equations in algebras, for propositional formulas, for open formulas of predicate logic, and for the formulas of quantifier logic. The structural description of logical consequence is a straightforward extension of that of equational consequence, as long as Boolean valued propositions and Boolean valued structures are considered; the reduction of the classical 2-valued case then depends on the Boolean prime ideal theorem.
Sets Models And Proofs
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Author : Ieke Moerdijk
language : en
Publisher: Springer
Release Date : 2018-11-23
Sets Models And Proofs written by Ieke Moerdijk and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-23 with Mathematics categories.
This textbook provides a concise and self-contained introduction to mathematical logic, with a focus on the fundamental topics in first-order logic and model theory. Including examples from several areas of mathematics (algebra, linear algebra and analysis), the book illustrates the relevance and usefulness of logic in the study of these subject areas. The authors start with an exposition of set theory and the axiom of choice as used in everyday mathematics. Proceeding at a gentle pace, they go on to present some of the first important results in model theory, followed by a careful exposition of Gentzen-style natural deduction and a detailed proof of Gödel’s completeness theorem for first-order logic. The book then explores the formal axiom system of Zermelo and Fraenkel before concluding with an extensive list of suggestions for further study. The present volume is primarily aimed at mathematics students who are already familiar with basic analysis, algebra and linear algebra. It contains numerous exercises of varying difficulty and can be used for self-study, though it is ideally suited as a text for a one-semester university course in the second or third year.
Set Theoretical Logic The Algebra Of Models
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Author : W Felscher
language : en
Publisher: CRC Press
Release Date : 2000-05-30
Set Theoretical Logic The Algebra Of Models written by W Felscher and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-05-30 with Mathematics categories.
This is an introduction to mathematical logic in which all the usual topics are presented: compactness and axiomatizability of semantical consequence, Löwenheim-Skolem-Tarski theorems, prenex and other normal forms, and characterizations of elementary classes with the help of ultraproducts. Logic is based exclusively on semantics: truth and satisfiability of formulas in structures are the basic notions. The methods are algebraic in the sense that notions such as homomorphisms and congruence relations are applied throughout in order to gain new insights. These concepts are developed and can be viewed as a first course on universal algebra. The approach to algorithms generating semantical consequences is algebraic as well: for equations in algebras, for propositional formulas, for open formulas of predicate logic, and for the formulas of quantifier logic. The structural description of logical consequence is a straightforward extension of that of equational consequence, as long as Boolean valued propositions and Boolean valued structures are considered; the reduction of the classical 2-valued case then depends on the Boolean prime ideal theorem.
Model Theory An Introduction
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Author : David Marker
language : en
Publisher: Springer Science & Business Media
Release Date : 2006-04-06
Model Theory An Introduction written by David Marker and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-04-06 with Mathematics categories.
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
Introduction To Model Theory
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Author : Philipp Rothmaler
language : en
Publisher: CRC Press
Release Date : 2000-10-31
Introduction To Model Theory written by Philipp Rothmaler and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-10-31 with Mathematics categories.
Model theory investigates mathematical structures by means of formal languages. So-called first-order languages have proved particularly useful in this respect. This text introduces the model theory of first-order logic, avoiding syntactical issues not too relevant to model theory. In this spirit, the compactness theorem is proved via the algebraically useful ultrsproduct technique (rather than via the completeness theorem of first-order logic). This leads fairly quickly to algebraic applications, like Malcev's local theorems of group theory and, after a little more preparation, to Hilbert's Nullstellensatz of field theory. Steinitz dimension theory for field extensions is obtained as a special case of a much more general model-theoretic treatment of strongly minimal theories. There is a final chapter on the models of the first-order theory of the integers as an abelian group. Both these topics appear here for the first time in a textbook at the introductory level, and are used to give hints to further reading and to recent developments in the field, such as stability (or classification) theory.
Introduction To Mathematical Logic
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Author : Jerome Malitz
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Introduction To Mathematical Logic written by Jerome Malitz and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
This book is intended as an undergraduate senior level or beginning graduate level text for mathematical logic. There are virtually no prere quisites, although a familiarity with notions encountered in a beginning course in abstract algebra such as groups, rings, and fields will be useful in providing some motivation for the topics in Part III. An attempt has been made to develop the beginning of each part slowly and then to gradually quicken the pace and the complexity of the material. Each part ends with a brief introduction to selected topics of current interest. The text is divided into three parts: one dealing with set theory, another with computable function theory, and the last with model theory. Part III relies heavily on the notation, concepts and results discussed in Part I and to some extent on Part II. Parts I and II are independent of each other, and each provides enough material for a one semester course. The exercises cover a wide range of difficulty with an emphasis on more routine problems in the earlier sections of each part in order to familiarize the reader with the new notions and methods. The more difficult exercises are accompanied by hints. In some cases significant theorems are devel oped step by step with hints in the problems. Such theorems are not used later in the sequence.
Mathematical Logic And Model Theory
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Author : Alexander Prestel
language : en
Publisher: Springer Science & Business Media
Release Date : 2011-08-21
Mathematical Logic And Model Theory written by Alexander Prestel and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2011-08-21 with Mathematics categories.
Mathematical Logic and Model Theory: A Brief Introduction offers a streamlined yet easy-to-read introduction to mathematical logic and basic model theory. It presents, in a self-contained manner, the essential aspects of model theory needed to understand model theoretic algebra. As a profound application of model theory in algebra, the last part of this book develops a complete proof of Ax and Kochen's work on Artin's conjecture about Diophantine properties of p-adic number fields. The character of model theoretic constructions and results differ quite significantly from that commonly found in algebra, by the treatment of formulae as mathematical objects. It is therefore indispensable to first become familiar with the problems and methods of mathematical logic. Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic treatment of several algebraic theories (Chapter 4). This book will be of interest to both advanced undergraduate and graduate students studying model theory and its applications to algebra. It may also be used for self-study.
Nonstandard Models Of Arithmetic And Set Theory
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Author : Ali Enayat
language : en
Publisher: American Mathematical Soc.
Release Date : 2004
Nonstandard Models Of Arithmetic And Set Theory written by Ali Enayat and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.
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.
Model Theoretic Logics
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Author : J. Barwise
language : en
Publisher: Cambridge University Press
Release Date : 2017-03-02
Model Theoretic Logics written by J. Barwise and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-02 with Mathematics categories.
This book brings together several directions of work in model theory between the late 1950s and early 1980s.
An Algebraic Introduction To Mathematical Logic
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Author : D.W. Barnes
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-06-29
An Algebraic Introduction To Mathematical Logic written by D.W. Barnes and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-06-29 with Mathematics categories.
This book is intended for mathematicians. Its origins lie in a course of lectures given by an algebraist to a class which had just completed a substantial course on abstract algebra. Consequently, our treatment of the subject is algebraic. Although we assume a reasonable level of sophistication in algebra, the text requires little more than the basic notions of group, ring, module, etc. A more detailed knowledge of algebra is required for some of the exercises. We also assume a familiarity with the main ideas of set theory, including cardinal numbers and Zorn's Lemma. In this book, we carry out a mathematical study of the logic used in mathematics. We do this by constructing a mathematical model of logic and applying mathematics to analyse the properties of the model. We therefore regard all our existing knowledge of mathematics as being applicable to the analysis of the model, and in particular we accept set theory as part of the meta-Ianguage. We are not attempting to construct a foundation on which all mathematics is to be based--rather, any conclusions to be drawn about the foundations of mathematics come only by analogy with the model, and are to be regarded in much the same way as the conclusions drawn from any scientific theory.