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Title: Computability and Logic by George S. Boolos, John P. Burgess, Richard C. Jeffrey ISBN: 0-521-00758-5 Publisher: Cambridge University Press Pub. Date: March, 2002 Format: Paperback Volumes: 1 List Price(USD): $28.00 |
Average Customer Rating: 3.79 (14 reviews)
Rating: 5
Summary: Very lucid explanations
Comment: This book is regarded as a 'classic' and rightly so. It assumes a minimal background, some familiarity with the propositional calculus. Even this can be dispensed with, if the reader is sufficiently motivated, as there is a well-written review of the first-order logic that one typically learns in an introductory formal logic course.
The book is highly readable. Each chapter begins with a short paragraph outlining the topics in the chapter, how they relate to each other, and how they connect with the topics in later and earlier chapters. These intros by themselves are valuable. The explanations though are what stand out. The authors are somehow able to take the reader's hand and guide him/her leisurely along with plentiful examples, but without getting bogged down in excessive prose. And they are somehow able to cover a substantive amount of material in a short space without seeming rushed or making the text too dense. It's nothing short of miraculous.
What made the book especially appealing to me is that it starts right out with Turing Machines. As a topologist who recently got interested in computational topology, I needed a book that would quickly impart a good, intuitive grasp of the basic notions of computability. I have more "mathematical maturity" than is needed to read an introductory book on computability, so I feel confident in saying that most of the standard texts on computability revel in excessive detail, like defining Turing Machines as a 6-tuple -- something that serves no purpose other than pedantry. This book is different. I particularly liked how the authors stress the intuitive notions underlying the definitions. For example, they lay special emphasis on the Church-Turing thesis, always asking the reader to consider how arguments can be simplified if it were true.
One should note that the emphasis of this book is more towards logic. While it starts with issues of computability, it moves into issues of provability, consistency, etc. The book covers the standards such as Goedel's famous incompleteness theorems in addition to some less standard topics at the end of the book. A small set of instructive exercises follows each chapter.
Rating: 5
Summary: Rigorous but readable
Comment: Just about the only mathematical logic book readable by ordinary human beings, assuming only a background of first-order predicate logic, taught in almost any introductory logic class.
The first eight chapters introduce Turing machines and other formal models of computation, emphazing the evidence for Church's thesis. Chapters 9-13 prove important results concerning first-order logic, including soundness, completeness, compactness, and Lowenheim-Skolem. The rest of the book focuses on number theory, and proves results such as Goedel's incompleteness theorems, Loeb's theorem, the existence and structure of non-standard models of arithmetic, and the decidability of Presburger arithmetic.
Highly recommended as an introduction to mathematical logic.
Rating: 3
Summary: almost great
Comment: Except for the scores of typos. Previous reviewers have observed this already; one has added that Burgess maintains an errata file on his website at Princeton. In fact he has two (for 1st and 2nd printings). But note that the errata file, at least for the 1st edition, is far from complete. I've noticed at least a dozen (potentially very confusing) typos that he has not yet catalogued. It's very frustrating to have to check the errata file (over 40 pages!) everytime one gets confused.
Two more points (1) the proof of compactness could have been better organized, and thereby made less tedious. (2) In general, there could stand to be more meta-level discussion about what's going on in the book. I find it's mostly trees, very little forest. (I'm not asking for _Godel, Escher, Bach_ here; I mean: where is this proof headed? Where did these satisfacton properties come from? etc)
On the positive side, the book is comprehensive, with very little handwaving, and the chapters are usually short and sweet. I prefer this text to Mendelson's. Enderton's is not bad.
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Title: Mathematical Logic by Joseph R. Shoenfield ISBN: 1568811357 Publisher: AK Peters, Ltd. Pub. Date: 15 January, 2001 List Price(USD): $39.00 |
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Title: Introduction to Mathematical Logic, Fourth Edition by Elliot Mendelson, Elliott Mendelson ISBN: 0412808307 Publisher: Lewis Publishers, Inc. Pub. Date: 01 June, 1997 List Price(USD): $79.95 |
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Title: Godel's Proof by Ernest Nagel, James Roy Newman, Douglas R. Hofstadter ISBN: 0814758169 Publisher: New York University Press Pub. Date: February, 2002 List Price(USD): $18.95 |
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Title: An Introduction to Non-Classical Logic by Graham Priest ISBN: 052179434X Publisher: Cambridge University Press Pub. Date: 22 February, 2001 List Price(USD): $20.00 |
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Title: A New Introduction to Modal Logic by G. E. Hughes, M. J. Cresswell ISBN: 0415126002 Publisher: Routledge Pub. Date: October, 1996 List Price(USD): $25.99 |
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