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Mathematical Logic

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

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

Table of contents

  1. Front Matter
    Pages i-x
  2. Part A

    1. Front Matter
      Pages 1-1
    2. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 3-9
    3. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 11-25
    4. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 27-57
    5. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 59-74
    6. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 75-85
    7. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 87-98
    8. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 99-114
    9. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 115-133
  3. Part B

    1. Front Matter
      Pages 135-135
    2. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 137-149
    3. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 151-187
    4. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 189-241
    5. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 243-259
    6. H.-D. Ebbinghaus, J. Flum, W. Thomas
      Pages 261-276
  4. Back Matter
    Pages 277-290

About this book

Introduction

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.

Keywords

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

Authors and affiliations

  • H.-D. Ebbinghaus
    • 1
  • J. Flum
    • 1
  • W. Thomas
    • 2
  1. 1.Mathematisches InstitutUniversität FreiburgFreiburgGermany
  2. 2.Institut für Informatik und Praktische MathematikUniversität KielKielGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4757-2355-7
  • Copyright Information Springer-Verlag New York 1994
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4757-2357-1
  • Online ISBN 978-1-4757-2355-7
  • Series Print ISSN 0172-6056
  • Buy this book on publisher's site
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