Ebook: A Course on Mathematical Logic

This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, recursion theory, and computability. In this new edition, many small and large changes have been made throughout the text. The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic. Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry. Some proofs, such as the proof of the very important completeness theorem, have been completely rewritten in a more clear and concise manner. The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more.

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This is a short, modern, and motivated introduction to mathematical logic for upper undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in getting acquainted with logic and would like to learn Gödel’s incompleteness theorems should find this book particularly useful. The treatment is thoroughly mathematical and prepares students to branch out in several areas of mathematics related to foundations and computability, such as logic, axiomatic set theory, model theory, recursion theory, and computability. In this new edition, many small and large changes have been made throughout the text. The main purpose of this new edition is to provide a healthy first introduction to model theory, which is a very important branch of logic. Topics in the new chapter include ultraproduct of models, elimination of quantifiers, types, applications of types to model theory, and applications to algebra, number theory and geometry. Some proofs, such as the proof of the very important completeness theorem, have been completely rewritten in a more clear and concise manner. The new edition also introduces new topics, such as the notion of elementary class of structures, elementary diagrams, partial elementary maps, homogeneous structures, definability, and many more. Table of Contents Cover A Course on Mathematical Logic, Second Edition ISBN 9781461457459 ISBN 9781461457466 Preface Contents Chapter 1 Syntax of First-Order Logic 1.1 First-Order Languages 1.2 Terms of a Language 1.3 Formulas of a Language 1.4 First-Order Theories Chapter 2 Semantics of First-Order Languages 2.1 Structures of First-Order Languages 2.2 Truth in a Structure 2.3 Models and Elementary Classes 2.4 Embeddings and Isomorphisms 2.5 Some Examples 2.6 Homogeneous Structures 2.7 Downward L�wenheim-Skolem Theorem 2.8 Definability Chapter 3 Propositional Logic 3.1 Syntax of Propositional Logic 3.2 Semantics of Propositional Logic 3.3 Compactness Theorem for Propositional Logic 3.4 Proof in Propositional Logic 3.5 Metatheorems in Propositional Logic 3.6 Post Tautology Theorem Chapter 4 Completeness Theorem for First-Order Logic 4.1 Proof in First-Order Logic 4.2 Metatheorems in First-Order Logic 4.3 Consistency and Completeness 4.4 Proof of the Completeness Theorem 4.5 Interpretations in a Theory 4.6 Extension by Definitions 4.7 Some Metatheorems in Arithmetic Chapter 5 Model Theory 5.1 Applications of the Completeness Theorem 5.2 Compactness Theorem 5.3 Upward L�wenheim-Skolem Theorem 5.4 Ultraproduct of Models 5.5 Some Applications in Algebra 5.6 Extensions of Partial Elementary Maps 5.7 Elimination of Quanti ers 5.8 Applications of Elimination of Quanti ers 5.9 Real Closed Fields 5.10 Some Applications in Algebra and Geometry 5.11 Isolated and Omitting Types 5.12 Relative Types 5.13 Prime and Atomic Models 5.14 Saturated Models 5.15 Stable Theories Chapter 6 Recursive Functions and Arithmetization of Theories 6.1 Recursive Functions and Recursive Predicates 6.2 Semirecursive Predicates 6.3 Arithmetization of Theories 6.4 Decidable Theories Chapter 7 Representability and Incompleteness Theorems 7.1 Representability 7.2 First Incompleteness Theorem 7.3 Arithmetical Sets 7.4 Recursive Extensions of Peano Arithmetic 7.5 Second Incompleteness Theorem References Index