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Computability and logic

Many books on mathematical logic soon plunge the reader into a mass of difficult to understand symbols. Computability and logic by Boolos, Burgess and Jeffrey starts off in an easier to follow style, and so gives the reader a gentler learning curve for the subject. The book starts off with a look at Turing machines, showing how they can be considered equivalent to any other computer. This is followed by an introduction to the theory of recursive functions. The second part of the book deals with first order logic, leading to the proof of Gödel's incompleteness theorem and related results.

The third part of the book moves on to more advanced topics such as second order logic and nonstandard models.

The authors generally don't give full proofs of theorems, but rather give a sketch which will enable the readers to complete the proof for themselves. This not only enables more material to be covered, but is done in a way that I felt enhanced the readers' understanding of the topics, and so the book is useful as a reference to the important considerations in the proofs. The book is aimed at those studying mathematical logic at undergraduate level, but I felt that it would also be useful for those wanting to undertake independent study of the subject.

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Paperback 368 pages
ISBN: 0521007585
Salesrank: 876597
Weight:1.5 lbs

Published: 2002 Cambridge University Press

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Paperback 368 pages
ISBN: 0521007585
Salesrank: 809730
Weight:1.5 lbs

Published: 2002 Cambridge University Press

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Paperback 368 pages
ISBN: 0521007585
Salesrank: 510746
Weight:1.5 lbs

Published: 2002 Cambridge University Press

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Product Description
Now in its fourth edition, this book has become a classic because of its accessibility to students without a mathematical background, and because it covers not only the staple topics of an intermediate logic course such as Godel's Incompleteness Theorems, but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has enhanced the book by adding a selection of problems at the end of each chapter.

Product Description

Now in its fourth edition, this book has become a classic because of its accessibility to students without a mathematical background, and because it covers not only the staple topics of an intermediate logic course such as Godel's Incompleteness Theorems, but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has enhanced the book by adding a selection of problems at the end of each chapter.

Without doubt the one to go for *****
This is a classic and an absolute must for anyone required (or wanting) to gain insight into intermediate logic. A more accessible (and yet more comprehensive) introduction is simply not available. The first part introduces basic concepts of computation, the second goes through the standard stock of important first-order result (culminating, of course, in the incompleteness theorems) whereas the third part goes through various further topics, including the Interpolation theorem (obviously), nonstandard models and provability (especially Loeb's theorem). And the style? The philosophical lexicon contains the following entry: [boo, n. The length of a mathematical or logical proof; hence, booloss, n., the process of shortening such a proof. "Only after significant booloss could the compactness theorem be explained in fifteen minutes."]. That one is pretty apt. With the Berry paradox, Boolos is able to prove the first incompleteness theorem in approximately half a page (a more standard approach is of course included as well). He has elsewhere explained the second incompleteness theorem using only one-syllable words. Point is: these authors (and, one suspects, Boolos in particular) has (had) an almost scary ability to make difficult things simple and easily comprehensible. That said, I do have a few misgivings. The typos have of course been mentioned (a list of errata is available on http://www.princeton.edu/~jburgess/addenda.htm), but most have been corrected in the second printing (so if you buy the book new, you'll probably get this one) - there are a few left, however. I am also not sure about some of the changes to the fourth edition. In particular, the structure of the proofs of the completeness, compactness and Löwenheim-Skolem theorems is somewhat surprising, proceeding from two lemmas concerning "satisfaction properties" and "closure properties". It is an interesting move, but will (partially because of the presentation, admittedly) surely be somewhat confusing to anyone coming to these for the first time not already being aware of how they fit together. That said, there is no way I can give this book less than five stars. There is simply no relevant competition comparable in accessibility and comprehensiveness. Urgently recommended.

Without doubt the one to go for *****

This is a classic and an absolute must for anyone required (or wanting) to gain insight into intermediate logic. A more accessible (and yet more comprehensive) introduction is simply not available. The first part introduces basic concepts of computation, the second goes through the standard stock of important first-order result (culminating, of course, in the incompleteness theorems) whereas the third part goes through various further topics, including the Interpolation theorem (obviously), nonstandard models and provability (especially Loeb's theorem).

And the style? The philosophical lexicon contains the following entry: [boo, n. The length of a mathematical or logical proof; hence, booloss, n., the process of shortening such a proof. "Only after significant booloss could the compactness theorem be explained in fifteen minutes."]. That one is pretty apt. With the Berry paradox, Boolos is able to prove the first incompleteness theorem in approximately half a page (a more standard approach is of course included as well). He has elsewhere explained the second incompleteness theorem using only one-syllable words. Point is: these authors (and, one suspects, Boolos in particular) has (had) an almost scary ability to make difficult things simple and easily comprehensible.

That said, I do have a few misgivings. The typos have of course been mentioned (a list of errata is available on http://www.princeton.edu/~jburgess/addenda.htm), but most have been corrected in the second printing (so if you buy the book new, you'll probably get this one) - there are a few left, however. I am also not sure about some of the changes to the fourth edition. In particular, the structure of the proofs of the completeness, compactness and Löwenheim-Skolem theorems is somewhat surprising, proceeding from two lemmas concerning "satisfaction properties" and "closure properties". It is an interesting move, but will (partially because of the presentation, admittedly) surely be somewhat confusing to anyone coming to these for the first time not already being aware of how they fit together.

That said, there is no way I can give this book less than five stars. There is simply no relevant competition comparable in accessibility and comprehensiveness. Urgently recommended.

One step from greatness. ****
The main virtue of this book, and which sets it apart from most other modern textbooks I have seen, is that it provides clear and usually illuminating explanations of the philosophical importance of the topics covered. These explanations and clarifications are given in a clear and usually crisp prose and emphasise the philosophical importance of whatever metalogical method or result they concern. I regard it as a very suitable companion or reference-work for the philosophically interested student of logic. For rigorous and very detailed proofs and definitions I normally consult a book like Mendelson's Introduction to Mathematical Logic, but usually I read what Boolos, Burgess and Jeffrey say too. The fact that the book (in its fourth edition at least) is divided into many short chapters makes it all the more useful as a companion. The short 'abstracts' that introduce each chapter deserve special mention. An index is the best way of localising information about something one knows one needs. The abstracts often do the reverse; they help one realise what one needs. As other reviewers have pointed out, the book has WAY TOO MANY typos. Burgess has a list of errata on his web-page, but it is not exhaustive and above all a professionally edited book should not have this many typos. The typos is in my mind the only thing that prevents it from earning five stars.

One step from greatness. ****

The main virtue of this book, and which sets it apart from most other modern textbooks I have seen, is that it provides clear and usually illuminating explanations of the philosophical importance of the topics covered. These explanations and clarifications are given in a clear and usually crisp prose and emphasise the philosophical importance of whatever metalogical method or result they concern. I regard it as a very suitable companion or reference-work for the philosophically interested student of logic. For rigorous and very detailed proofs and definitions I normally consult a book like Mendelson's Introduction to Mathematical Logic, but usually I read what Boolos, Burgess and Jeffrey say too. The fact that the book (in its fourth edition at least) is divided into many short chapters makes it all the more useful as a companion. The short 'abstracts' that introduce each chapter deserve special mention. An index is the best way of localising information about something one knows one needs. The abstracts often do the reverse; they help one realise what one needs.
As other reviewers have pointed out, the book has WAY TOO MANY typos. Burgess has a list of errata on his web-page, but it is not exhaustive and above all a professionally edited book should not have this many typos. The typos is in my mind the only thing that prevents it from earning five stars.

Math majors avoid **
If you are a math major, you don't want this book. Get Cutland. This book was written by philosophy professors and shows it. When philosophers write math, it is less concise, organized, and complete than when mathematicians do it. This was meant as an intermediate logic text for philosophy and math students, but it would try the patience of a math major. The explanations are wordy, sketchy, and poorly related to each other and to exterior topics. The contents fall into thirds: Turing machines, aspects of decidability, and a hodge podge of topics from other parts of logic. The 4e is 50 pages (17%) longer than the 3e. The changes were mainly adding exercises and making the chapters more independent. The authors were obviously trying hard to make a readable text, but I hated slogging through all that verbiage. You can see a lot of comment in the other reviews that the 3e was better, but I think even the 3e is poor compared to Cutland.

Math majors avoid **

If you are a math major, you don't want this book. Get Cutland.

This book was written by philosophy professors and shows it. When philosophers write math, it is less concise, organized, and complete than when mathematicians do it. This was meant as an intermediate logic text for philosophy and math students, but it would try the patience of a math major. The explanations are wordy, sketchy, and poorly related to each other and to exterior topics.

The contents fall into thirds: Turing machines, aspects of decidability, and a hodge podge of topics from other parts of logic. The 4e is 50 pages (17%) longer than the 3e. The changes were mainly adding exercises and making the chapters more independent. The authors were obviously trying hard to make a readable text, but I hated slogging through all that verbiage. You can see a lot of comment in the other reviews that the 3e was better, but I think even the 3e is poor compared to Cutland.

Absolutely rediculous *
WAY TOO MANY TYPOS!!!!!! There were so many typos, it made it extremely difficult to follow this book at times. As a first time student to mathematical logic, I found this to be just too much. People who are veterans with logic and logicians may easily spot typos, but for a first time student of the subject, I was confused as hell at some parts simply because there was a typo. I wasted hours trying to figure out some parts (such as the factorial function in chapter 6) when I finally found out that the reason why I couldn't figure it out was because of a typo. The Errata sheet on the internet IS 35 PAGES LONG!!!! I didn't pay money to correct a horde of typos! God that pisses me off.

Absolutely rediculous *

WAY TOO MANY TYPOS!!!!!! There were so many typos, it made it extremely difficult to follow this book at times. As a first time student to mathematical logic, I found this to be just too much. People who are veterans with logic and logicians may easily spot typos, but for a first time student of the subject, I was confused as hell at some parts simply because there was a typo. I wasted hours trying to figure out some parts (such as the factorial function in chapter 6) when I finally found out that the reason why I couldn't figure it out was because of a typo. The Errata sheet on the internet IS 35 PAGES LONG!!!! I didn't pay money to correct a horde of typos! God that pisses me off.

Prefer the old edition ***
I have used the old edition for a class in computability and logic where the students did not have much background in either. Having used the new edition this year, I find I greatly prefer the old one. The new one may be more rigorous, but it is much harder to read and understand for students without the background. The first part is not so bad, but the second half on logic gets too involved in the proofs and the students lose sight of the overall pupose and what these result really mean.

Prefer the old edition ***

I have used the old edition for a class in computability and logic where the students did not have much background in either. Having used the new edition this year, I find I greatly prefer the old one. The new one may be more rigorous, but it is much harder to read and understand for students without the background. The first part is not so bad, but the second half on logic gets too involved in the proofs and the students lose sight of the overall pupose and what these result really mean.

One step from greatness ****
The main virtue of this book, and which sets it apart from most other modern textbooks I have seen, is that it provides clear and usually illuminating explanations of the philosophical importance of the topics covered. These explanations and clarifications are given in a clear and usually crisp prose and emphasise the philosophical importance of whatever metalogical method or result they concern. I regard it as a very suitable companion or reference-work for the philosophically interested student of logic. For rigorous and very detailed proofs and definitions I normally consult a book like Mendelson's Introduction to Mathematical Logic, but usually I read what Boolos, Burgess and Jeffrey say too. The fact that the book (in its fourth edition at least) is divided into many short chapters makes it all the more useful as a companion. The short 'abstracts' that introduce each chapter deserve special mention. An index is the best way of localising information about something one knows one needs. The abstracts often do the reverse; they help one realise what one needs. There are, however, WAY TOO MANY typos. Burgess has a list of errata on his web-page, but it is not exhaustive and above all a professionally produced book should not have this many typos. This is, in my mind, the only thing that prevents it from earning five stars.

One step from greatness ****

The main virtue of this book, and which sets it apart from most other modern textbooks I have seen, is that it provides clear and usually illuminating explanations of the philosophical importance of the topics covered. These explanations and clarifications are given in a clear and usually crisp prose and emphasise the philosophical importance of whatever metalogical method or result they concern. I regard it as a very suitable companion or reference-work for the philosophically interested student of logic. For rigorous and very detailed proofs and definitions I normally consult a book like Mendelson's Introduction to Mathematical Logic, but usually I read what Boolos, Burgess and Jeffrey say too. The fact that the book (in its fourth edition at least) is divided into many short chapters makes it all the more useful as a companion. The short 'abstracts' that introduce each chapter deserve special mention. An index is the best way of localising information about something one knows one needs. The abstracts often do the reverse; they help one realise what one needs.
There are, however, WAY TOO MANY typos. Burgess has a list of errata on his web-page, but it is not exhaustive and above all a professionally produced book should not have this many typos. This is, in my mind, the only thing that prevents it from earning five stars.

This is a standard work. *****
This is simply to say that this is a marvellous and, apart from anything else, standard work, and that therefore the only other review presented by Amazon grossly distorts its nature and status. A must buy!

This is a standard work. *****

This is simply to say that this is a marvellous and, apart from anything else, standard work, and that therefore the only other review presented by Amazon grossly distorts its nature and status. A must buy!

Not for the faint-hearted **
This book is FULL of information, the hard part is knowing how to understand it. This was the recommended book for my BSc degree and I found it SO hard to understand. The terminology and diagrams were vague and ambiguous. Unless you attend Oxford or Cambridge, don't bother with this book.

Not for the faint-hearted **

This book is FULL of information, the hard part is knowing how to understand it. This was the recommended book for my BSc degree and I found it SO hard to understand. The terminology and diagrams were vague and ambiguous. Unless you attend Oxford or Cambridge, don't bother with this book.

Not much to add, but ***
it should be noted that this book is not intended for the auto-didact. Like other good logic texts-Jeffrey's Formal Logic or Pollock's Technical Method's (out of print, but available in PDF on his website)-there is very little commentary in the brief chapters, so it is useful if you are already very familiar with the material or if you have a very worthy guide. An advantage of the short chapters is that material is broken down in finer increments; a disadvantage is that material is presented with spare guidance at times. I was also disappointed by the sparsity of examples. Like many logic and math students, I learn better from examining a few examples than I do from either lectures or text: give me three examples of something and I'll usually have it down. I would have liked to see more examples in this text. The exercises are ample and creative, which I appreciate, but often go so far beyond the text it's mind boggling. They often require extensive extrapolations from the text sometimes even proving theorems or lemmas not in the text just for use in the exercise. I should say that I'm a philosopher and not a mathematician (I suspect the other reviewers are primarily mathematicians), so my estimation of the difficulty will differ. I aced Symbolic Logic, Modal Logic, Deviant Logic, and Advanced Symbolic Logic and still had difficulty with some of this material, even though I had a prior acquaintance with Godel's proof. Note that the first reviewer, who thought it was a breeze, described himself this way "As a topologist who recently got interested in computational topology..." Good for him, but if you are not a professional mathematician this book will probably be quite challenging at times, even if you are otherwise good at mathematical logic. Note also that the second five-star review refers to the older edition-it has not necessarily improved with age. I firmly agree with the reviewer from Brooklyn that the proofs could have had more forecasting and with the reviewer from Raleigh that a solution set, say to the odds, would have been very useful, especially for the auto-didact, from whose perspective I am writing.

Not much to add, but ***

it should be noted that this book is not intended for the auto-didact. Like other good logic texts-Jeffrey's Formal Logic or Pollock's Technical Method's (out of print, but available in PDF on his website)-there is very little commentary in the brief chapters, so it is useful if you are already very familiar with the material or if you have a very worthy guide. An advantage of the short chapters is that material is broken down in finer increments; a disadvantage is that material is presented with spare guidance at times. I was also disappointed by the sparsity of examples. Like many logic and math students, I learn better from examining a few examples than I do from either lectures or text: give me three examples of something and I'll usually have it down. I would have liked to see more examples in this text. The exercises are ample and creative, which I appreciate, but often go so far beyond the text it's mind boggling. They often require extensive extrapolations from the text sometimes even proving theorems or lemmas not in the text just for use in the exercise. I should say that I'm a philosopher and not a mathematician (I suspect the other reviewers are primarily mathematicians), so my estimation of the difficulty will differ. I aced Symbolic Logic, Modal Logic, Deviant Logic, and Advanced Symbolic Logic and still had difficulty with some of this material, even though I had a prior acquaintance with Godel's proof. Note that the first reviewer, who thought it was a breeze, described himself this way "As a topologist who recently got interested in computational topology..." Good for him, but if you are not a professional mathematician this book will probably be quite challenging at times, even if you are otherwise good at mathematical logic. Note also that the second five-star review refers to the older edition-it has not necessarily improved with age. I firmly agree with the reviewer from Brooklyn that the proofs could have had more forecasting and with the reviewer from Raleigh that a solution set, say to the odds, would have been very useful, especially for the auto-didact, from whose perspective I am writing.

Great textbook, poor text ****
I can hardly imagine a better introduction to the topics covered than this book. It discusses virtually everything the intermediate logic student could want: diagonalization, Turing machines, undeciability, indefinability, incompleteness, forcing, and on and on. Although the first few chapters are a bit awkward, the style is generally crystal clear and the examples and metaphors vivid. It's far and away the best read of any text on logic I've yet encountered. As a mathematician, I was concerned about the books' emphasis on logic rather than mathematics (the text is aimed at philosophy students, too). But the introduction to foundations flows so easily and naturally that I could never complain. Anyone interested in the topic, regardless of their background, could hardly do better (or cheaper) for an introduction. P.S. - I wanted to give this five stars, but, as other reviewers have pointed out, there are simply too many typos. C'mon, get an editor.

Great textbook, poor text ****

I can hardly imagine a better introduction to the topics covered than this book. It discusses virtually everything the intermediate logic student could want: diagonalization, Turing machines, undeciability, indefinability, incompleteness, forcing, and on and on. Although the first few chapters are a bit awkward, the style is generally crystal clear and the examples and metaphors vivid. It's far and away the best read of any text on logic I've yet encountered.

As a mathematician, I was concerned about the books' emphasis on logic rather than mathematics (the text is aimed at philosophy students, too). But the introduction to foundations flows so easily and naturally that I could never complain. Anyone interested in the topic, regardless of their background, could hardly do better (or cheaper) for an introduction.

P.S. - I wanted to give this five stars, but, as other reviewers have pointed out, there are simply too many typos. C'mon, get an editor.

almost great ***
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.

almost great ***

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.

Nice Revision ***
This 4th edition text requires much diligence and patience, even given the author's clarity and excellent presentation. The problem sets are very useful to readers/scholars of most levels. Some answers (not just hints) would be helpful, though. The most damning feature of the book is its typos and errors. One of the authors (John Burgess) has an errata sheet online, but it is hardly reasonable for thousands of individuals to review these corrections and make them in thousands of texts when a competent editorial staff could have done the work.

Nice Revision ***

This 4th edition text requires much diligence and patience, even given the author's clarity and excellent presentation. The problem sets are very useful to readers/scholars of most levels. Some answers (not just hints) would be helpful, though.

The most damning feature of the book is its typos and errors. One of the authors (John Burgess) has an errata sheet online, but it is hardly reasonable for thousands of individuals to review these corrections and make them in thousands of texts when a competent editorial staff could have done the work.

Excellent update to third edition *****
This is the standard text for those who have only had an introductory logic class and want to work up to (and past) Godel's incompleteness theorems. The third edition was already good, and John Burgess has extensively rewritten parts to make the arguments clearer and easier to follow. It is true that there are a number of typos, but a list of corrections can be downloaded from John Burgess's web site at Princeton University.

Excellent update to third edition *****

This is the standard text for those who have only had an introductory logic class and want to work up to (and past) Godel's incompleteness theorems. The third edition was already good, and John Burgess has extensively rewritten parts to make the arguments clearer and easier to follow.

It is true that there are a number of typos, but a list of corrections can be downloaded from John Burgess's web site at Princeton University.

George S Boolos,John P Burgess and Richard C Jeffrey

Computability and logic