Mathematical Logic

1. Front Matter

   Pages i-x

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2. Part A

   1. Front Matter

      Pages 1-1

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   2. Introduction

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 3-9

   3. Syntax of First-Order Languages

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 11-25

   4. Semantics of First-Order Languages

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 27-57

   5. A Sequent Calculus

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 59-74

   6. The Completeness Theorem

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 75-85

   7. The Löwenheim-Skolem Theorem and the Compactness Theorem

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 87-98

   8. The Scope of First-Order Logic

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 99-114

   9. Syntactic Interpretations and Normal Forms

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 115-133

3. Part B

   1. Front Matter

      Pages 135-135

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   2. Extensions of First-Order Logic

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 137-149

   3. Limitations of the Formal Method

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 151-187

   4. Free Models and Logic Programming

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 189-241

   5. An Algebraic Characterization of Elementary Equivalence

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 243-259

   6. Lindström’s Theorems

      • H.-D. Ebbinghaus, J. Flum, W. Thomas

      Pages 261-276

4. Back Matter

   Pages 277-290

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What is a mathematical proof? How can proofs be justified? Are there limitations to provability? To what extent can machines carry out mathe­ matical proofs? Only in this century has there been success in obtaining substantial and satisfactory answers. The present book contains a systematic discussion of these results. The investigations are centered around first-order logic. Our first goal is Godel's completeness theorem, which shows that the con­ sequence relation coincides with formal provability: By means of a calcu­ lus consisting of simple formal inference rules, one can obtain all conse­ quences of a given axiom system (and in particular, imitate all mathemat­ ical proofs). A short digression into model theory will help us to analyze the expres­ sive power of the first-order language, and it will turn out that there are certain deficiencies. For example, the first-order language does not allow the formulation of an adequate axiom system for arithmetic or analysis. On the other hand, this difficulty can be overcome--even in the framework of first-order logic-by developing mathematics in set-theoretic terms. We explain the prerequisites from set theory necessary for this purpose and then treat the subtle relation between logic and set theory in a thorough manner.

• Arithmetic
• Equivalence
• Logic
• Mathematische Logik
• compactness theorem
• mathematical logic
• model theory
• proof

“…the book remains my text of choice for this type of material, and I highly recommend it to anyone teaching a first logic course at this level.” – Journal of Symbolic Logic

• Mathematisches Institut, Universität Freiburg, Freiburg, Germany

  H.-D. Ebbinghaus, J. Flum

• Institut für Informatik und Praktische Mathematik, Universität Kiel, Kiel, Germany

  W. Thomas

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