Customer Reviews

An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy)

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For a couple of decades now, students who had completed their first logic class and dabbled in a little bit of metatheory (perhaps soundness and completeness) were forced to avail themselves of Boolos and Jeffrey's (fourth edition with Burgess) "Computability and Logic." Unfortunately, the third edition presented much of the material in too brief a manner, resulting in a big jump from lower level logic to the material covered. The fourth edition is much longer, but no more easier to teach to talented undergraduates. More recently, Epstein's book on computability was an improvement in this regard, but its logical coverage was much less.

Smith's book should now be the canonical text. First, the discussion and proofs are astoundingly clear to students who haven't done much logic beyond their first class. Pick any topic from B & J and Smith, for example primitive recursiveness, the tie between p.r. axiomatizability and axiomatizability via Craig's theorem, etc. and the discussion and proofs in Smith will be clearer, more accessible, and more clearly tied to the other relevant concepts. Second, the coverage is exactly what is needed to understand both theorems and the most important consequences and extensions. Third, the way he ties the disparate topics together (for example the informal proofs through Chapter 5 and their rigorizaiton through Chapter 18) is just fantastic. This is really important for helping the reader develop a deeper understanding of things. If you just pile theorem upon theorem it's easy for the reader accept them as true without developing any logical insight and appreciation of the landscape.

I don't know if Cambridge would allow this, but in the next edition they should seriously think about adding exercise sections like B & J and Epstein. If they did, I think this book would eclipse the other two for classroom uses.

It's not just for students, either. A colleague and I were arguing about something and we picked up Smith's work rathern than either of Smullyan's to figure out a point relevant to the debate. I find that my grasp of the relevant proofs is much cleaer for reading Smith (my colleague is much, much better at logic than me, but with Smith's help I won the debate).

It is both extraordinary and a cause for celebration when someone can combine in a logic text this level of coverage, rigor, accessibility, and funness of read. I don't think there is a precedent actually. In short, Smith's work is a service to Lady Philosophy. Joe Bob says check it out.

This is a terrific book that the reader can learn a lot from. The author presents Gödel's theorems - in fact, he provides many different proofs of the theorems - along with various strenghtenings and weakenings of the main results. In the many historical and conceptual asides the author does a great job of explaining the significance of Gödel's theorems and of directing the reader's attention to the big picture.

However, I don't think the book is a good *introduction* to Gödel's theorems. A student approaching these theorems for the first time will be overwhelmed by the amount of information here. Even more problematic is the author's adoption of a rather informal way of writing. This does make the book very readable but I think would frustrate the beginning student who needs a precise grasp of new concepts. For example, I don't think a student innocent of primitive recursive functions would be able to grasp how they work from the chapter here. This problem is further compounded by the lack of exercises.

In sum, the book is highly recommended for anyone looking to deepen and broaden their understanding of Gödel's theorems. However, I think that anyone who hasn't already seen a rigorous presentation of those theorems might find the book frustrating.

Mathematical logic in all its dexterity is a somewhat fractured discipline. We have set theory, recursion theory, proof theory, model theory, etc. all falling under the same umbrella. Each of these disciplines move into very sophisticated terrain quickly, sometimes with disparate notation, techniques, goals, etc. Thus it can be a hurdle for one to try to get an overview of each area and their pivotal questions and theorems, despite the claim of their relation to "Mathematical Logic."

The incompleteness theorems of Godel have been an impetus for logical research since their discovery. So it is not out of place for a writer to want to focus on them and give a (relatively) uninitiated audience a survey of relevant results. What is wonderful about Peter Smith's text is how he presents the essentials clearly and touches on how the separate research areas have tried to deal with or explain incompleteness. So in this sense, Smith illustrates at least one way all the areas of mathematical logic are connected, namely how they all have at some point sought to shine light on Godel's results. The fruits of these areas' labors are made clear and accessible in a way that is not duplicated anywhere else; at present, this book is one of a kind.

The only flaw is the lack of exercises, but these can be found in more specialized texts. There is excellent annotation and an accompanying bibliography for those who want to go further.

This is absolutely the best text to go on to after you've already taken a basic course in mathematical logic. (Having worked my way through Boolos & Jeffrey, Bell & Machover, Mendelson, Kunen, and Shoenfield, among others, I can say this with some assurance.) I've read it twice, each time over the space of several months, and I've never felt more clear on the fine points of Gödel's theorems, recursion theory, and decidability. What I can't stress highly enough is how painless it is. Peter Smith has a real gift for exposition, and he's evidently gone through a lot of trouble to make the reasoning, terminology, and proofs as lucid as possible. And, needless to say, the content is intellectually thrilling. (For self-study, I would recommend starting with Geoffrey Hunter's "Metalogic," which is the easiest intro to mathematical logic I've come across. For side reading in various topics in logic, John Stillwell's "Roads to Infinity" is terrific.)

All the reviewers are enthusiastic and with good reason.

This is not just one more text on Gödel's theorems, nor just another standard source... : it's "THE" book.

Peter Smith is gifted both as a teacher and as a writer. He managed the "tour de force" of making Gödel's ideas accessible, clearly and uncompromisingly.

Whether it be global architecture, introduction and linkage of basic concepts, footnotes, other sources, text layout... Everything is well thought of , producing a sense of inner beauty.

This book should be read by any would-be writer in mathematics...

If you are really interested in the Gödelian world, buy this book and embark with Peter Smith on a splendid journey.

Very rigurous but simultaneously very understandable (...for this kind of theorems that have many technical dificulties is much to say...) I enjoy (very much) to read this book (I took it to the beach during my vacations!.)

I have read much of the 2007 edition and have found it very illuminating and helpful. However, Peter Smith says on his website that there now exists a fourth 2009 reprinting with many significant corrections and additions. Amazon's website description of the book mentions only the first printing. I have asked Amazon to confirm that the book it is now selling is this fourth corrected reprinting, but, alas, without success. Does anyone know whether Amazon is currently selling the fourth reprinting? An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy)