Mathematical Logic, 3/ED.

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ABOUT THE BOOK

This textbook introduces first-order logic and its role in the foundations of mathematics by examining fundamental questions. What is a mathematical proof? How can mathematical proofs be justified? Are there limitations to provability? To what extent can machines carry out mathematical proofs? In answering these questions, this textbook explores the capabilities and limitations of algorithms and proof methods in mathematics and computer science.

The chapters are carefully organized, featuring complete proofs and numerous examples throughout. Beginning with motivating examples, the book goes on to present the syntax and semantics of first-order logic. After providing a sequent calculus for this logic, a Henkin-type proof of the completeness theorem is given. These introductory chapters prepare the reader for the advanced topics that follow, such as Gödel's Incompleteness Theorems, Trakhtenbrot's undecidability theorem, Lindström's theorems on the maximality of first-order logic, and results linking logic with automata theory. This new edition features many modernizations, as well as two additional important results: The decidability of Presburger arithmetic, and the decidability of the weak monadic theory of the successor function.

Mathematical Logic is ideal for students beginning their studies in logic and the foundations of mathematics. Although the primary audience for this textbook will be graduate students or advanced undergraduates in mathematics or computer science, in fact the book has few formal prerequisites. It demands of the reader only mathematical maturity and experience with basic abstract structures, such as those encountered in discrete mathematics or algebra.


TABLE OF CONTENTS

  1. Part A

    1. Front Matter

      Pages 1-1
    2. Introduction

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 3-9
    3. Syntax of First-Order Languages

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 11-24
    4. Semantics of First-Order Languages

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 25-54
    5. A Sequent Calculus

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 55-70
    6. The Completeness Theorem

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 71-81
    7. The Löwenheim–Skolem Theorem and the Compactness Theorem

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 83-94
    8. The Scope of First-Order Logic

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 95-109
    9. Syntactic Interpretations and Normal Forms

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 111-129
  2. Part A

    1. Front Matter

      Pages 131-131
    2. Extensions of First-Order Logic

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 133-145
    3. Computability and Its Limitations

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 147-204
    4. Free Models and Logic Programming

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 205-256
    5. An Algebraic Characterization of Elementary Equivalence

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 257-272
    6. Lindström’s Theorems

      • Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas
      Pages 273-289

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