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Ernest Nagel and James R Newman

Godel's proof

Gödel's incompleteness theorem has a reputation of being mysterious and difficult to understand. In Gödel's Proof Ernest Nagel and James R Newman give a clear explanation of the basics of the proof. They explain the background to the proof and in particular the search for a way of proving consistency of a system of axioms. They then exhibit a system where such a proof is possible - the propostional calculus. The book continues with chapters on the concept of mapping in mathematics, and on Gödel numbering, finishing off with the proof of the incompleteness theorem itself.

I did have some misgivings about the book. I felt that some examples of proofs from axioms would have helped the reader to understand what was going on. Also the book doesn't mention Gödel's β function, which I consider to be the most innovative part of the proof - how you code for 'and so on..' when your system doesn't include such a concept. In summary, the book would be useful for those wishing to get a clear idea of Gödel's theorem without getting too technical, and to go further than what is given in most popular treatments of the subject, but it doesn't go that much further than such treatments.

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Paperback 112 pages
ISBN: 0415355281
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Published: 2005 Routledge

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Paperback 112 pages
ISBN: 0415355281
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Published: 2005 Routledge

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Paperback 112 pages
ISBN: 0415355281
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Product Description
In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences—perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times." However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject. New York University Press is proud to publish this special edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

Product Description

In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences—perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times."

However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.

New York University Press is proud to publish this special edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

Godel for people such as we (who are familiar with a little Theory of Numbrets) *****
I ran into Godel back in about 1955 in "Scientific American". I did not understand that. Now 53 years later, and with some more understanding of the theory of numbers, I find this work to be a magnificent opening into Godel's world. AS WOMDERFUL read!.

Godel for people such as we (who are familiar with a little Theory of Numbrets) *****

I ran into Godel back in about 1955 in "Scientific American". I did not understand that. Now 53 years later, and with some more understanding of the theory of numbers, I find this work to be a magnificent opening into Godel's world. AS WOMDERFUL read!.

A Simple Presentation of a Complex Topic *****
Godel's proof represents a milestone in mathematical and philosophical thought. This book, annotated by the remarkable Douglas Hofstadter, presents Godel's ideas in a language that's readily understandable by the educated layperson. It is clear, concise, and fascinating.

A Simple Presentation of a Complex Topic *****

Godel's proof represents a milestone in mathematical and philosophical thought. This book, annotated by the remarkable Douglas Hofstadter, presents Godel's ideas in a language that's readily understandable by the educated layperson. It is clear, concise, and fascinating.

Godel's incompleteness theorems explained in non-technical language *****
There is no question in my mind that the most misunderstood mathematical theorems of all time are Godel's incompleteness theorems. In essence, they state that no system powerful enough to do basic arithmetic is complete. Meaning that there will always be statements that are true in the system but that can never be proven in that system. These results have been seized upon by people with many different agendas and used to argue conclusions as significant as the existence of God and that human intelligence is not simply the outward manifestation of neurotransmitters flowing from place to place. While this is all somewhat amusing, it is also disquieting, as the theorems cannot be used to conclusively justify such significant claims. This book is one of the very first books where an attempt is made to explain Godel's theorems to the mathematical laity. In that sense, it is a success; the appropriate background is effectively put forward before the theorem and proof are explained. There is little in the way of formal mathematics and the bulk of the terminology is non-technical. If the people who use Godel's results to justify their extravagant claims were to read this book with an open mind, they would recognize the absurdity of their positions.

Godel's incompleteness theorems explained in non-technical language *****

There is no question in my mind that the most misunderstood mathematical theorems of all time are Godel's incompleteness theorems. In essence, they state that no system powerful enough to do basic arithmetic is complete. Meaning that there will always be statements that are true in the system but that can never be proven in that system. These results have been seized upon by people with many different agendas and used to argue conclusions as significant as the existence of God and that human intelligence is not simply the outward manifestation of neurotransmitters flowing from place to place. While this is all somewhat amusing, it is also disquieting, as the theorems cannot be used to conclusively justify such significant claims.
This book is one of the very first books where an attempt is made to explain Godel's theorems to the mathematical laity. In that sense, it is a success; the appropriate background is effectively put forward before the theorem and proof are explained. There is little in the way of formal mathematics and the bulk of the terminology is non-technical. If the people who use Godel's results to justify their extravagant claims were to read this book with an open mind, they would recognize the absurdity of their positions.

Godel's incompleteness theorem, clearly explained ****
Gödel's incompleteness theorem remains one of the most quoted, yet most misunderstood work in mathematics of the last century. Many non mathematicians had used the theorem (without understanding it, of course), to "prove" just about everything (generally it's been used to imply the limits of science, or stuff to that effect). The theorem, though, hardly implies that. This short book, written some 50 years ago, remains probably the best explanation of Godel's work available to the layman. The book starts explaining the background to Godel's theorem, as mathematicians such as Hilbert and Russell sought the axiomatization of mathematics. Godel's work, of course, proved that to be impossible. The book then proceeds to explain the theorem itself, as clearly as it possibly can (though I have to say that, as a non expert, the Gödel numbering scheme seems a like a trick to me, a sleight of hand. Yet, what do I know about this?). Overall, a great book about a much misunderstood work.

Godel's incompleteness theorem, clearly explained ****

Gödel's incompleteness theorem remains one of the most quoted, yet most misunderstood work in mathematics of the last century. Many non mathematicians had used the theorem (without understanding it, of course), to "prove" just about everything (generally it's been used to imply the limits of science, or stuff to that effect). The theorem, though, hardly implies that. This short book, written some 50 years ago, remains probably the best explanation of Godel's work available to the layman. The book starts explaining the background to Godel's theorem, as mathematicians such as Hilbert and Russell sought the axiomatization of mathematics. Godel's work, of course, proved that to be impossible. The book then proceeds to explain the theorem itself, as clearly as it possibly can (though I have to say that, as a non expert, the Gödel numbering scheme seems a like a trick to me, a sleight of hand. Yet, what do I know about this?). Overall, a great book about a much misunderstood work.

how i understand Godel *****
Godel was able to construct a formula from the axioms of Principia Mathematica (PM, and related systems, due to Russell and Whitehead) that is (roughly) "There exists no proof for this formula". This IS the formula itself. now, we need to know if this is true or not. so we try to find a proof for it within PM. if it is possible to find the proof, that means the formula is correct. but the formula says you can't find it. if the proof cannot be found, that means the negation of the formula is correct, but then the formula tells you that you cannot find it, so the formula must be correct, not its negation. in another words, the formula is undecidable within PM. also meaning PM is incomplete. what is amazing now is, of course that is what the formula tells you. so we do know it is true eventhough we can't show it within PM. note that some people like reviewer Paul Vjecsner, who has also posted his disagreements on other Godel-related books, still confuse mathematics and meta-mathematics. although i have described and mixed-up meta-mathematics meanings to the formula above, Godel's proof was completely mathematicized within PM. either Vjecsner didn't understand the proof or he underestimated the grand aim of PM and thus the significance of Godel's work in taking it apart. Vjecsner argues that Godel's proof is essentially a linguistic paradox that has been unreasonably translated into PM. but the proof doesn't need that. that explanation is only done to give the readers a vague glimpse of the proof in a meta-mathemtical level. the proof only shows that there is a formula which is decidable if its negation is decidable. if you then argue that this is an unacceptable formula, then you are making even bigger claim than Godel, namely that the system is inconsistent! but the problem for you is you can't prove that within PM, thus still showing that PM is incomplete. Godel's proof is only as meaningful as PM. if you poke hole at Godel, you are poking hole at PM, which is exactly what Godel wanted to prove. the book gives you an outline of how Godel went about constructing that formula with the language of PM, how he made the proof number-theoretical, and many more details. of course reading Godel's original paper would still be nightmarishly difficult even for many mathematicians, so the book gives a very good 'Godel's proof for dummies'. so you want to know what that formula looks like? read this book.

how i understand Godel *****

Godel was able to construct a formula from the axioms of Principia Mathematica (PM, and related systems, due to Russell and Whitehead) that is (roughly) "There exists no proof for this formula". This IS the formula itself. now, we need to know if this is true or not. so we try to find a proof for it within PM. if it is possible to find the proof, that means the formula is correct. but the formula says you can't find it. if the proof cannot be found, that means the negation of the formula is correct, but then the formula tells you that you cannot find it, so the formula must be correct, not its negation. in another words, the formula is undecidable within PM. also meaning PM is incomplete. what is amazing now is, of course that is what the formula tells you. so we do know it is true eventhough we can't show it within PM.

note that some people like reviewer Paul Vjecsner, who has also posted his disagreements on other Godel-related books, still confuse mathematics and meta-mathematics. although i have described and mixed-up meta-mathematics meanings to the formula above, Godel's proof was completely mathematicized within PM. either Vjecsner didn't understand the proof or he underestimated the grand aim of PM and thus the significance of Godel's work in taking it apart. Vjecsner argues that Godel's proof is essentially a linguistic paradox that has been unreasonably translated into PM. but the proof doesn't need that. that explanation is only done to give the readers a vague glimpse of the proof in a meta-mathemtical level. the proof only shows that there is a formula which is decidable if its negation is decidable. if you then argue that this is an unacceptable formula, then you are making even bigger claim than Godel, namely that the system is inconsistent! but the problem for you is you can't prove that within PM, thus still showing that PM is incomplete. Godel's proof is only as meaningful as PM. if you poke hole at Godel, you are poking hole at PM, which is exactly what Godel wanted to prove.

the book gives you an outline of how Godel went about constructing that formula with the language of PM, how he made the proof number-theoretical, and many more details. of course reading Godel's original paper would still be nightmarishly difficult even for many mathematicians, so the book gives a very good 'Godel's proof for dummies'. so you want to know what that formula looks like? read this book.

Nagel and Newman *****
This book by Nagel and Newman is a great classic, clever and suited to the intellectually curious who have not the patience (or talent) for the full syntax of mathematical logic. Aside: It is a little misleading that Amazon, in their header, say "By Douglas R.Hofstadter, ..." as he wrote the preface only to this much later reprinted edition.

Nagel and Newman *****

This book by Nagel and Newman is a great classic, clever and suited to the intellectually curious who have not the patience (or talent) for the full syntax of mathematical logic.

Aside: It is a little misleading that Amazon, in their header, say "By Douglas R.Hofstadter, ..." as he wrote the preface only to this much later reprinted edition.

The best introduction for the most revolutionary proof in Logic. *****
"Gödel's Proof" is a classic. Despite of being brief it doesn't sacrifice punctuality for length. There are no superfluous paragraphs in this book which leads the leader with steady but not lengthy steps to the understanding of the greatest leap taken in the field of Mathematical Logic. The text is not written in a technical style and even the few mathematical proofs found at the last chapters are indispensable for understanding the heard of Gödel's Proof. Don't be confused from the mathematical title of this book and think that this book is written only for mathematicians. On the contrary I believe that the importance of this proof reaches every natural science because the language used from the majority of the these sciences is mathematics and therefore the characteristics of the language used to describe natural processes affects the epistemological foundations of the aforementioned natural sciences. This unique textbook has also a lot to do with cognitive science and the way we think as human beings while trying to understand our natural environment. For this reason I consider "Gödel's Proof" as a wonderful book of cognitive science too. I truly recommend this book to anyone who want to take a deep dive in the big blue of what mathematics are and their connection with empirical reality. It is the best book to get a good feeling about Mathematical Logic and reasoning.

The best introduction for the most revolutionary proof in Logic. *****

"Gödel's Proof" is a classic. Despite of being brief it doesn't sacrifice punctuality for length. There are no superfluous paragraphs in this book which leads the leader with steady but not lengthy steps to the understanding of the greatest leap taken in the field of Mathematical Logic. The text is not written in a technical style and even the few mathematical proofs found at the last chapters are indispensable for understanding the heard of Gödel's Proof.

Don't be confused from the mathematical title of this book and think that this book is written only for mathematicians. On the contrary I believe that the importance of this proof reaches every natural science because the language used from the majority of the these sciences is mathematics and therefore the characteristics of the language used to describe natural processes affects the epistemological foundations of the aforementioned natural sciences.

This unique textbook has also a lot to do with cognitive science and the way we think as human beings while trying to understand our natural environment. For this reason I consider "Gödel's Proof" as a wonderful book of cognitive science too. I truly recommend this book to anyone who want to take a deep dive in the big blue of what mathematics are and their connection with empirical reality. It is the best book to get a good feeling about Mathematical Logic and reasoning.

Outstanding introductory text *****
For those interested, but uninitiated, in the philosophy of mathematics or mathematical philosophy should seriously consider reading this excellent introductory text. In a highly concise and lucid manner, the authors successfully explain the origins, development and details of Godel's proof and examine some of the wider implication of it. It is not, however, particularly easy reading. Unlike reading a novel, it requires some effort to fully understand and grapple with the strange but intriguing concepts discussed. No background or logic necessary; technicalities are generally avoided. All in all, outstanding. Well worth buying.

Outstanding introductory text *****

For those interested, but uninitiated, in the philosophy of mathematics or mathematical philosophy should seriously consider reading this excellent introductory text. In a highly concise and lucid manner, the authors successfully explain the origins, development and details of Godel's proof and examine some of the wider implication of it.

It is not, however, particularly easy reading. Unlike reading a novel, it requires some effort to fully understand and grapple with the strange but intriguing concepts discussed. No background or logic necessary; technicalities are generally avoided.

All in all, outstanding. Well worth buying.

An excellent guide to Gödel *****
Simply magnificent. This book meets and exceeds the description on its back cover -- offering "any educated person with a taste for logic and philosophy the chance to satisfy his intellectual curiosity about a previously inaccessible subject." This book gives anyone with the interest and the motivation a solid, if not complete, understanding of the ideas underlying the proof. While it's true that someone very unfamiliar with mathematics (or, more importantly, with logic and mathematical thinking) would not get as much out of the book, it does a very good job of walking the reader through Gödel's complex but breathtakingly elegant reasoning. I wholeheartedly recommend this book.

An excellent guide to Gödel *****

Simply magnificent. This book meets and exceeds the description on its back cover -- offering "any educated person with a taste for logic and philosophy the chance to satisfy his intellectual curiosity about a previously inaccessible subject." This book gives anyone with the interest and the motivation a solid, if not complete, understanding of the ideas underlying the proof. While it's true that someone very unfamiliar with mathematics (or, more importantly, with logic and mathematical thinking) would not get as much out of the book, it does a very good job of walking the reader through Gödel's complex but breathtakingly elegant reasoning. I wholeheartedly recommend this book.

A Very Good Introduction ****
I Read this book in an afternoon. While this book covers many of Godels ideas, concepts, and systematically works throught the incompleteness theory, it does however lack the fine detail of the actual theorem. I recomend this book for those who wish to find out aout Godels Proof, without wanting the know the fine details.

A Very Good Introduction ****

I Read this book in an afternoon. While this book covers many of Godels ideas, concepts, and systematically works throught the incompleteness theory, it does however lack the fine detail of the actual theorem. I recomend this book for those who wish to find out aout Godels Proof, without wanting the know the fine details.

Good attempt to explain the proof ****
This was clearly one of the best attempts at explaining Godel's proof that I have seen, at least superficially speaking. As someone who just wanted to understand what the basic ideas are, I looked over various books and decided on this one because of its high rating. I gave it 4 stars because I was left feeling that there were several times when background knowledge of higher mathematics/logic was assumed and I think more could have been done to explain those parts on a level comprehensible to an interested layperson. I think the attempt in the book is a good one, but I guess perhaps not enough is said about just how abstract these ideas are and how difficult it is to simply dive in (even with a good book) and expect to understand this proof fully. I am going to try Godel, Escher, Bach, and Roger Penrose's Shadows of the Mind next, since I have heard that both of them also include explanations of Godel's theorem. But I now have a greater appreciation of why there will never be a "Godel's Proof for Dummies" book!

Good attempt to explain the proof ****

This was clearly one of the best attempts at explaining Godel's proof that I have seen, at least superficially speaking. As someone who just wanted to understand what the basic ideas are, I looked over various books and decided on this one because of its high rating. I gave it 4 stars because I was left feeling that there were several times when background knowledge of higher mathematics/logic was assumed and I think more could have been done to explain those parts on a level comprehensible to an interested layperson.

I think the attempt in the book is a good one, but I guess perhaps not enough is said about just how abstract these ideas are and how difficult it is to simply dive in (even with a good book) and expect to understand this proof fully.

I am going to try Godel, Escher, Bach, and Roger Penrose's Shadows of the Mind next, since I have heard that both of them also include explanations of Godel's theorem. But I now have a greater appreciation of why there will never be a "Godel's Proof for Dummies" book!

Lucid & satisfying: Godel's Proof and modern logic *****
In 100 lucid and highly readable pages, presents the most important ideas of modern logic: axiomatisation (Euclid), formalization (Hilbert), metamathematical argumentation, consistency, completeness, etc., leading up to Godel's incompleteness result. Elementary from a technical point of view, but technical people should read it to get perspective. Non-technical people will appreciate its workmanlike, substantive exposition, in contrast to the mysticism, obfuscation, and cuteness of a "Godel, Escher, Bach". It is old (1958) and very incomplete (no set theory, no computability, no non-standard analysis, ...), but still essential reading. (I wrote this review in 1998, but Amazon doesn't know I'm the same person as macrakis@alum.mit.edu.)

Lucid & satisfying: Godel's Proof and modern logic *****

In 100 lucid and highly readable pages, presents the most important ideas of modern logic: axiomatisation (Euclid), formalization (Hilbert), metamathematical argumentation, consistency, completeness, etc., leading up to Godel's incompleteness result. Elementary from a technical point of view, but technical people should read it to get perspective. Non-technical people will appreciate its workmanlike, substantive exposition, in contrast to the mysticism, obfuscation, and cuteness of a "Godel, Escher, Bach". It is old (1958) and very incomplete (no set theory, no computability, no non-standard analysis, ...), but still essential reading.

(I wrote this review in 1998, but Amazon doesn't know I'm the same person as macrakis@alum.mit.edu.)

Don't be intimidated by the subject matter. *****
The greatest merit of this book is its ability to take a rather arcaic and complicated proof and successfully present it, in a concise and understandable manner, to a broad audience. An otherwise motivated and intelligent person with almost no background in logic should enjoy and understand most of Nagel and Jackson's summarization. One technique that Nagel and Jackson employ is to repeat themselves, presenting crucial points in two or three slightly different ways to insure the idea is grasped. The short length not only makes a one night read a possibility, but makes it easier to grasp the broad structure of the proof itself.

Don't be intimidated by the subject matter. *****

The greatest merit of this book is its ability to take a rather arcaic and complicated proof and successfully present it, in a concise and understandable manner, to a broad audience. An otherwise motivated and intelligent person with almost no background in logic should enjoy and understand most of Nagel and Jackson's summarization. One technique that Nagel and Jackson employ is to repeat themselves, presenting crucial points in two or three slightly different ways to insure the idea is grasped. The short length not only makes a one night read a possibility, but makes it easier to grasp the broad structure of the proof itself.

Wish I'd read it first ... *****
I read Godel's paper in grad school. I wish I had read this first, because it lays out the structure of the argument clearly. N&N are particularly good on clarifying what Godel did and did not prove. This is important because of all the loose mystical obfuscation out there about this theorem. N&N clearly explain what formal "games with marks" methods are, and why mathematicians resort to them. They then walk through what Godel proved, with a bit on how he proved it. The basic idea of his (blitheringly complex) mapping is explained quite well indeed. Suitable for mathematicians, or philosophy students tired of mystical speculations. Also goo for anyone with an interest in computability theory or any formal logic. And read it before you read Godel's paper!

Wish I'd read it first ... *****

I read Godel's paper in grad school. I wish I had read this first, because it lays out the structure of the argument clearly. N&N are particularly good on clarifying what Godel did and did not prove. This is important because of all the loose mystical obfuscation out there about this theorem.

N&N clearly explain what formal "games with marks" methods are, and why mathematicians resort to them. They then walk through what Godel proved, with a bit on how he proved it. The basic idea of his (blitheringly complex) mapping is explained quite well indeed.

Suitable for mathematicians, or philosophy students tired of mystical speculations. Also goo for anyone with an interest in computability theory or any formal logic. And read it before you read Godel's paper!

A Must Read for Math and Philosophy Students *****
Any mathematician or philosopher who has an interest in the foundations of mathematics should be familiar with Godel's work. A mathematician reading GP may long for a more rigorous accounting of Godel's proof but GP is still an excellent exegesis because of how nicely it paints Godel's theorem in broad strokes. A more technical account can be found in Smullyan's book on Godel's Theorem, which is published by Oxford. Lazy philosophers and laypeople will appreciate this book and should definitely purchase and read it before delving into a more complicated account of Godel's incompleteness theorems. In sum, this book is clearly written and probably the most elementary introduction to Godel's theorems out there. As for those of you reading this review and wondering just what's important about Godel's theorem, here are some of its highlights: 1) Godel's work shows us that there are definite limits to formal systems. Just because we can formulate a statement within a formal system doesn't mean we can derive it or make sense of it without ascending to a metalevel. (Just a note: Godel's famous statement which roughly translates as "I am not provable" is comprehensible only from the metalevel. It corresponds to a statement that can be formed in the calculus but not derived in it, if we assume the calculus to be correct.) 2) Godel's famous sentence represents an instance of something referring to itself indirectly. 3) Godel's method of approaching the problem is novel in that he found a way for sentences to talk about themselves within a formal system. 4) His proof shows to be incorrect the belief that if we just state mathematical problems clearly enough we will find a solution. Godel's theory is somewhat esoteric; there just aren't that many math and philosophy majors out there and there are even fewer people who have a relatively solid grasp of the proof, even at a macro level. If you want to learn about one of the most interesting and impressive intellectual achievements of the 20th century, I highly recommend you get this book.

A Must Read for Math and Philosophy Students *****

Any mathematician or philosopher who has an interest in the foundations of mathematics should be familiar with Godel's work.

A mathematician reading GP may long for a more rigorous accounting of Godel's proof but GP is still an excellent exegesis because of how nicely it paints Godel's theorem in broad strokes. A more technical account can be found in Smullyan's book on Godel's Theorem, which is published by Oxford.

Lazy philosophers and laypeople will appreciate this book and should definitely purchase and read it before delving into a more complicated account of Godel's incompleteness theorems.

In sum, this book is clearly written and probably the most elementary introduction to Godel's theorems out there.

As for those of you reading this review and wondering just what's important about Godel's theorem, here are some of its highlights:

1) Godel's work shows us that there are definite limits to formal systems. Just because we can formulate a statement within a formal system doesn't mean we can derive it or make sense of it without ascending to a metalevel. (Just a note: Godel's famous statement which roughly translates as "I am not provable" is comprehensible only from the metalevel. It corresponds to a statement that can be formed in the calculus but not derived in it, if we assume the calculus to be correct.)

2) Godel's famous sentence represents an instance of something referring to itself indirectly.

3) Godel's method of approaching the problem is novel in that he found a way for sentences to talk about themselves within a formal system.

4) His proof shows to be incorrect the belief that if we just state mathematical problems clearly enough we will find a solution.

Godel's theory is somewhat esoteric; there just aren't that many math and philosophy majors out there and there are even fewer people who have a relatively solid grasp of the proof, even at a macro level. If you want to learn about one of the most interesting and impressive intellectual achievements of the 20th century, I highly recommend you get this book.