Overview
- Authors:
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H.-D. Ebbinghaus
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Mathematisches Institut, Universität Freiburg, Freiburg, Germany
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J. Flum
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Mathematisches Institut, Universität Freiburg, Freiburg, Germany
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W. Thomas
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Institut für Informatik und Praktische Mathematik, Universität Kiel, Kiel, Germany
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Table of contents (13 chapters)
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Part A
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 3-9
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 11-25
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 27-57
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 59-74
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 75-85
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 87-98
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 99-114
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 115-133
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Part B
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Front Matter
Pages 135-135
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 137-149
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 151-187
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 189-241
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 243-259
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- H.-D. Ebbinghaus, J. Flum, W. Thomas
Pages 261-276
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Back Matter
Pages 277-290
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