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

Peter Smith (Author)

Book Description

Publication Date: August 6, 2007 | ISBN-10: 0521674530 | ISBN-13: 978-0521674539 | Edition: 1

In 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter?  Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic.

Editorial Reviews

Review

"How did Gödel establish the two Theorems of Incompleteness, and why do they matter? Smith (U. of Cambridge) advises readers to take their time in answering these and related questions he poses as he presents a variety of proofs for the First Theorem and shows how to prove the Second. He also examines a group of related results with the same care and attention to detail. In 36 well-paced chapters Smith builds his case from a basic introduction to G<:o>del's theorems on to such issues as the truths of arithmetic, formalized arithmetics, primitive recursive functions, identifying the diagonalization Lemma in the First Theorem and using it, dirivability conditions in the Second Theorem. Turing machines (and recursiveness) and the Church-Turing thesis. Accessible without being dismissive, this is accessible to philosophy students and equally suitable for mathematics students taking a first course in logic."
Book News

"... Without doubt, a mandatory reference for every philosopher interested in philosophy of mathematics. The text is, in general, written in a prose style but without avoiding formalisms. It is very accurate in the mathematical arguments and it offers to mathematicians and logicians a detailed approach to Gödel's theorems, covering many aspects which are not easy to find in other presentations."
Reinhard Kahle, Mathematical Reviews

Book Description

What are Gödel's Theorems, how were they established and why do they matter? Written with great clarity, this book is accessible to philosophy students with a limited formal background. It is equally valuable to mathematics students taking a first course in mathematical logic.

See all Editorial Reviews

Product Details

• Paperback: 376 pages
• Publisher: Cambridge University Press; 1 edition (August 6, 2007)
• Language: English
• ISBN-10: 0521674530
• ISBN-13: 978-0521674539
• Product Dimensions: 9.6 x 7 x 0.7 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (7 customer reviews)
• Amazon Best Sellers Rank: #427,391 in Books (See Top 100 in Books)

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44 of 44 people found the following review helpful:

5.0 out of 5 stars the best thing out there, December 25, 2007

By 

Jon M. Cogburn (Baton Rouge, LA.) - See all my reviews
(REAL NAME)   

This review is from: An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy) (Paperback)

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.

14 of 14 people found the following review helpful:

5.0 out of 5 stars Good Book, Wrong Title, January 11, 2009

By 

Pedestrian "W" (Saint Louis, United States) - See all my reviews

This review is from: An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy) (Paperback)

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.

2 of 2 people found the following review helpful:

5.0 out of 5 stars Gives some Research Directions, April 26, 2011

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Customer (Baltimore) - See all my reviews

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This review is from: An Introduction to Gödel's Theorems (Cambridge Introductions to Philosophy) (Paperback)

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.