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.
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'Smith has written a wonderful book giving a clear and compelling presentation of Gödel's Theorems and their implications. His style is both precise and engaging at the same time. The clarity of the writing is impressive, and there is a pleasing coverage of historical and philosophical topics. An Introduction to Gödel's Theorems will work very well either as a textbook or as an introduction for any reader who wants a thorough understanding of some of the central ideas at the intersection of philosophy, mathematics and computer science.' Christopher Leary, State University of New York
'Peter Smith has succeeded in writing an excellent introduction to Gödel's incompleteness theorems and related topics which is accessible without being superficial. Philosophers in particular will appreciate the discussions of the Church-Turing Thesis, mechanism, and the relevance of Gödel's results in the philosophy of mathematics. It is certain to become a standard text.' Richard Zach, University of Calgary
'… it is, 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 standard presentations.' Mathematical Reviews
'Peter Smith has succeeded in writing an excellent introduction to Gödel's incompleteness theorems and related topics which is accessible without being superficial. Philosophers in particular will appreciate the discussions of the Church-Turing Thesis, mechanism, and the relevance of Gödel's results in the philosophy of mathematics. It is certain to become a standard text.' Richard Zach, University of Calgary
'… it is, 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 standard presentations.' 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.
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22 of 22 people found the following review helpful
Format:Paperback
I recommend this book without reservation to anyone who wants to know about incompleteness, or about computability.
The main subjects of the book are as follows:
(1) It provides an excellent introduction to Godel's Theorems. A beginner in mathematical logic will easily be able to follow the introductory chapters, and thereby gain a good grasp of the incompleteness theorems. In particular, Smith gently introduces the technique of proof by diagonalisation, and shows how to use this to sketch a very easy and elegant incompleteness theorem).
(2) It provides systematic and detailed developments/reconstructions of Godel's proofs. In this sense, do not be fooled by the title's claim that this is an "Introduction"; it covers material far beyond introductory level. (Where it does so, this is clearly flagged.)
(3) It explains the relationship between incompleteness results and computability. Starting from scratch, the reader is introduced gently to Turing's Halting Problem, and then shown the link between this and the incompleteness results. Smith closes with an excellent philosophical commentary on the Church-Turing thesis; the rest of the book is lightly peppered with lucid philosophical commentary.
The single greatest achievement of this book is its clarity. Many books on mathematical logic present incredibly compressed proofs, and any commentary is likewise dry and terse. By contrast, Smith's writing style is cheerful and incredibly clear. He normally proceeds as follows: he first explains the proof-strategy in chatty, conversational English. This is followed by a full and technically rigorous proof, but written (again) in a manner that is easy to follow. He then closes with a simple summary, and a discussion of what the proof shows.... Read more ›
The main subjects of the book are as follows:
(1) It provides an excellent introduction to Godel's Theorems. A beginner in mathematical logic will easily be able to follow the introductory chapters, and thereby gain a good grasp of the incompleteness theorems. In particular, Smith gently introduces the technique of proof by diagonalisation, and shows how to use this to sketch a very easy and elegant incompleteness theorem).
(2) It provides systematic and detailed developments/reconstructions of Godel's proofs. In this sense, do not be fooled by the title's claim that this is an "Introduction"; it covers material far beyond introductory level. (Where it does so, this is clearly flagged.)
(3) It explains the relationship between incompleteness results and computability. Starting from scratch, the reader is introduced gently to Turing's Halting Problem, and then shown the link between this and the incompleteness results. Smith closes with an excellent philosophical commentary on the Church-Turing thesis; the rest of the book is lightly peppered with lucid philosophical commentary.
The single greatest achievement of this book is its clarity. Many books on mathematical logic present incredibly compressed proofs, and any commentary is likewise dry and terse. By contrast, Smith's writing style is cheerful and incredibly clear. He normally proceeds as follows: he first explains the proof-strategy in chatty, conversational English. This is followed by a full and technically rigorous proof, but written (again) in a manner that is easy to follow. He then closes with a simple summary, and a discussion of what the proof shows.... Read more ›
6 of 6 people found the following review helpful
By Enthusiast
Format:Paperback|Amazon Verified Purchase
This is a book written by someone who is clearly not only a master of the technical details of mathematical logic in the Godelian tradition, but is also well aware of the difficulties faced by the struggling student, who may be coming to grips with very complex ideas for the first time.
Most books on logic (or on mathematics generally for that matter) that achieve any level of technical difficulty are like a sat-nav system that tells you "turn right", "turn left" etc without explaining where you are actually going to, or why you are doing it. Smith's book by contrast is "readable" in the sense of carrying the reader along with constant explanations of where the text is going, and what is the point of the current proof. It motivates the student to keep going like no other text I have come across. I would say it is an essential companion volume for anyone who has bought Mendelson's famous book on mathematical logic, which scores highly on accuracy and completeness and very low on comprehensibility. Together they probably answer most questions a second level student will probably want to know.
Most books on logic (or on mathematics generally for that matter) that achieve any level of technical difficulty are like a sat-nav system that tells you "turn right", "turn left" etc without explaining where you are actually going to, or why you are doing it. Smith's book by contrast is "readable" in the sense of carrying the reader along with constant explanations of where the text is going, and what is the point of the current proof. It motivates the student to keep going like no other text I have come across. I would say it is an essential companion volume for anyone who has bought Mendelson's famous book on mathematical logic, which scores highly on accuracy and completeness and very low on comprehensibility. Together they probably answer most questions a second level student will probably want to know.
3 of 3 people found the following review helpful
Format:Paperback
I didn't have the chance to take a course on Gödel's theorems and I wanted to read something by myself. But soon I realized that there are two kind of texts: either too demanding for anyone with a modest background in logic and mathematics or too general (and trivial) for anyone with a modest background in logic and mathematics. This book is very well balanced in this respect: it gives you the details of the proofs most of the time (it sketches a proof or points out when to skip a proof when the thing is not really interesting) so that you get a real feeling of understanding the interesting ideas underlying the theorems; at the same time the author guides the reader with 'road-maps' when a proof is too long to retain all the details from the begining to the end. Also, there are interesting remarks on the philosphical impact of the results shown in the book.
I'm reading now chapter 21 and I've decided to read it till the end. I didn't follow in all detail some informal discussion concerning algorithms in the first chapters, but this is a minor thing for the results I've read so far. I definitely recommend it. Last thing: if you plan to buy it, try to get the latest version since, it seems, it has some corrections (otherwise there's a list of typos in the authors webpage).
I'm reading now chapter 21 and I've decided to read it till the end. I didn't follow in all detail some informal discussion concerning algorithms in the first chapters, but this is a minor thing for the results I've read so far. I definitely recommend it. Last thing: if you plan to buy it, try to get the latest version since, it seems, it has some corrections (otherwise there's a list of typos in the authors webpage).
1 of 1 people found the following review helpful
Format:Paperback
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.
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.
1 of 1 people found the following review helpful
By T. Rashid
Format:Paperback|Amazon Verified Purchase
I abandoned Godel's Theorem: An Incomplete Guide to Its Use and Abuse because the writing was terrible. In reviewing that book I complained that the topic itself wasn't necessarily difficult, but the writing style and inconsistency put up barriers.
This book is a pleasure to read, clear, structured, and gentle. I've only got ot chapter 3 but in comparison to the other one, this is working really well.
I'll update once I've read it all.
This book is a pleasure to read, clear, structured, and gentle. I've only got ot chapter 3 but in comparison to the other one, this is working really well.
I'll update once I've read it all.
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