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Mathematical Logic

  • Textbook
  • © 1994

Overview

Authors:
  1. H.-D. Ebbinghaus

    1. Mathematisches Institut, Universität Freiburg, Freiburg, Germany

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  2. J. Flum

    1. Mathematisches Institut, Universität Freiburg, Freiburg, Germany

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  3. W. Thomas

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

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

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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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Keywords

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.

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

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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