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Mathematical Logic

  • Textbook
  • © 1994

Overview

Authors:
  1. H.-D. Ebbinghaus

    1. Mathematisches Institut, Universität Freiburg, Freiburg, Germany

  2. J. Flum

    1. Mathematisches Institut, Universität Freiburg, Freiburg, Germany

  3. W. Thomas

    1. Institut für Informatik und Praktische Mathematik, Universität Kiel, Kiel, Germany

Part of the book series: Undergraduate Texts in Mathematics (UTM)

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About this book

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.

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Table of contents (13 chapters)

  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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Reviews

“…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

Authors and Affiliations

  • 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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Bibliographic Information

  • Book Title: Mathematical Logic

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

  • Series Title: Undergraduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4757-2355-7

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1994

  • Hardcover ISBN: 978-0-387-94258-2Published: 10 June 1994

  • eBook ISBN: 978-1-4757-2355-7Published: 14 March 2013

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 2

  • Number of Pages: X, 291

  • Topics: Mathematical Logic and Foundations, Mathematics Education

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