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Basic Proof Theory (Cambridge Tracts in Theoretical Computer Science) Paperback – July 31, 2000

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ISBN-13: 978-0521779111 ISBN-10: 0521779111 Edition: 2nd

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Basic Proof Theory (Cambridge Tracts in Theoretical Computer Science) + Model Theory: Third Edition (Dover Books on Mathematics)
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Editorial Reviews

Review

'This is a fine book. Any computer scientist with some logical background will benefit from studying it. It is written by two of the experts in the field and comes up to their usual standards of precision and care.' Ray Turner, Computer Journal

Book Description

This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included.
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Product Details

  • Series: Cambridge Tracts in Theoretical Computer Science (Book 43)
  • Paperback: 432 pages
  • Publisher: Cambridge University Press; 2 edition (July 31, 2000)
  • Language: English
  • ISBN-10: 0521779111
  • ISBN-13: 978-0521779111
  • Product Dimensions: 6 x 0.9 x 9 inches
  • Shipping Weight: 1.5 pounds (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Best Sellers Rank: #1,366,858 in Books (See Top 100 in Books)

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41 of 43 people found the following review helpful By William Stirton on March 18, 2005
Format: Paperback
This is a very bread-and-butter introduction to proof theory. Apart from digressions, it is not until we are five-sixths of the way through the book that we begin to meet formal systems in which any actual mathematics can be formalized (chapter 10). The first nine chapters are devoted to studying, in great detail, a plethora of purely logical systems. Anyone who thought, under the influence of Hilbert, perhaps, that proof theory was about proving the consistency of classical mathematics will probably be seriously disappointed with this book.

This is the main flaw in the book. Computer scientists (of whom I am not one) might like it; but beginners looking for an explanation of the relevance of proof theory to either mathematics or philosophy will probably not find what they are looking for, at least through the first five-sixths of the book.

Why is proof theory interesting? I could be missing something, but I just do not see that the authors have anything much to say about this question - rather a serious fault in an introductory textbook, surely? The book is very clear and the style is pleasant; but a great many hairs are split and a beginner cannot be expected to see that there is anything much to be gained from doing so.

Despite these faults, for readers who *already* possess a moderately advanced knowledge of proof theory and want a really thorough, in-depth treatment of the very basics of the subject, this book is very useful. A thing I particularly liked is the emphasis given to considerations about the lengths of proofs (sections 5.1 and 6.7). Some textbooks on proof theory either do not treat pure logic at all (Pohlers) or do treat it but without giving any information about what cut-elimination in pure logic does to the length of a proof (Schuette).
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By Jeffrey Rubard on December 2, 2003
Format: Paperback
[2015 review] I bought this book a few years after it was published, with Gentzen's "Investigations into Logical Deduction" under my belt; I could not make head or tail of most of the book, and gave up for a long while. In the meantime, I learned some actual computer science. Oh, what a difference a DFA makes! Troelstra and Schwichtenberg did not think interesting proof theory stops at cut-elimination, or at Gentzen's elaborate "proof" of the consistency of arithmetic using transfinite induction (Tarski claimed this latter item advanced his understanding of the issue "not one epsilon"). Modernized versions of these results are here, but the authors go into as much detail about resolution or category-theoretical logic or the proof theory of the simply typed lambda calculus, with its famous isomorphism to intuitionistic logic. Results of particular interest to computer scientists like the unmanageable complexity of cut-elimination algorithms, noted by George Boolos in "Don't Eliminate Cut", are stressed.

Proof theory is just beautiful compared to model theory and recursion theory, but knowing which way is up is as important as spilling "abstract nonsense". Gentzen himself is an excellent example of powerful insight rendered accessible to many, and though this book is not easy it isn't "intractable". All of the material will be of nearly bread-and-butter importance to the intended audience for this series, computer scientists with an interest in theory; and some stuff (like the proof theory of the modal logic S4 and the "translations" of seemingly incompatible logics one into another) is just fun.
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Basic Proof Theory (Cambridge Tracts in Theoretical Computer Science)
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