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ByJayon September 18, 2016

Overall, I very much enjoyed this book. It was as promised, a complete, though not thorough explanation of Godel's incompleteness theorem.

While it did use mathematical syntax throughout the book, it was all in the context of what was necessary for the explanation (and in many cases the mathematical symbols are easier to parse than the written explanation, however usually both are provided). Furthermore, tables are often shown throughout the book to further reinforce concepts.

As someone with an engineering rather than formal mathematics background, I found the book overall quite easy to follow. That is not to say that there weren't a few pages for which I needed to re-read several times, or that I would jump backwards in the book to reacquire the context of what was being explained, but it seemed like the book chose the right trade offs overall.

It is worth noting that the book is quite a quick read. I imagine most people would get through it in a little over a couple of hours. Yet the length of the book feels right. It manages to gloss over the right areas, while not affecting the subsequent explanations negatively.

While it did use mathematical syntax throughout the book, it was all in the context of what was necessary for the explanation (and in many cases the mathematical symbols are easier to parse than the written explanation, however usually both are provided). Furthermore, tables are often shown throughout the book to further reinforce concepts.

As someone with an engineering rather than formal mathematics background, I found the book overall quite easy to follow. That is not to say that there weren't a few pages for which I needed to re-read several times, or that I would jump backwards in the book to reacquire the context of what was being explained, but it seemed like the book chose the right trade offs overall.

It is worth noting that the book is quite a quick read. I imagine most people would get through it in a little over a couple of hours. Yet the length of the book feels right. It manages to gloss over the right areas, while not affecting the subsequent explanations negatively.

ByDennon June 17, 2016

I believe this book taught me something.... however it was not what I was expecting. I have read mathematics book in the past like "Prime Obsession," "Unknown Quantity," and "The Code Book" all being extremely interesting and making me die to continue reading. I was hoping this book would be like those. I found the authors, both mathematicians, to be extremely fact-oriented, often taking 4 sentences to explain something where a single sentence would have sufficed. The footnotes, taking 1/3 of the bottom of every other page, exacerbate the situation. I wanted a book that would capture my imagination; this book merely reinforced all the facts I can learn on Youtube with the exception of one (below). The mention of Godel's proof is merely 3 or 4 pages that vaguely review the summary Godel arrived at. This vague summary was not satisfying to me.

The reason I gave this 3 stars is because I learned about meta-mathematics. I also learned about systems where axioms are defined and how theorems in the system naturally follow from the axioms. Systems are everywhere, from playing chess to playing soccer and to driving. Fundamental rules are what keep these systems in flow. I liked how the authors dived into this a bit and because it is a short book I don't believe I wasted much time. The boring tone of this book reminded me of "GEB;" I suffered around 30 hours through GEB getting 70% done when I stopped reading, put the book away, and around 4 months later I gave it to a friend. I never heard from my friend or his thoughts on the book, I suppose he must have tossed it in the trash (which is fine with me).

The reason I gave this 3 stars is because I learned about meta-mathematics. I also learned about systems where axioms are defined and how theorems in the system naturally follow from the axioms. Systems are everywhere, from playing chess to playing soccer and to driving. Fundamental rules are what keep these systems in flow. I liked how the authors dived into this a bit and because it is a short book I don't believe I wasted much time. The boring tone of this book reminded me of "GEB;" I suffered around 30 hours through GEB getting 70% done when I stopped reading, put the book away, and around 4 months later I gave it to a friend. I never heard from my friend or his thoughts on the book, I suppose he must have tossed it in the trash (which is fine with me).

ByJayon September 18, 2016

Overall, I very much enjoyed this book. It was as promised, a complete, though not thorough explanation of Godel's incompleteness theorem.

While it did use mathematical syntax throughout the book, it was all in the context of what was necessary for the explanation (and in many cases the mathematical symbols are easier to parse than the written explanation, however usually both are provided). Furthermore, tables are often shown throughout the book to further reinforce concepts.

As someone with an engineering rather than formal mathematics background, I found the book overall quite easy to follow. That is not to say that there weren't a few pages for which I needed to re-read several times, or that I would jump backwards in the book to reacquire the context of what was being explained, but it seemed like the book chose the right trade offs overall.

It is worth noting that the book is quite a quick read. I imagine most people would get through it in a little over a couple of hours. Yet the length of the book feels right. It manages to gloss over the right areas, while not affecting the subsequent explanations negatively.

While it did use mathematical syntax throughout the book, it was all in the context of what was necessary for the explanation (and in many cases the mathematical symbols are easier to parse than the written explanation, however usually both are provided). Furthermore, tables are often shown throughout the book to further reinforce concepts.

As someone with an engineering rather than formal mathematics background, I found the book overall quite easy to follow. That is not to say that there weren't a few pages for which I needed to re-read several times, or that I would jump backwards in the book to reacquire the context of what was being explained, but it seemed like the book chose the right trade offs overall.

It is worth noting that the book is quite a quick read. I imagine most people would get through it in a little over a couple of hours. Yet the length of the book feels right. It manages to gloss over the right areas, while not affecting the subsequent explanations negatively.

ByDrewon January 2, 2014

This book is simple, short, and to the point. It explains the project of Principia Mathematica, the idea of building mathematics as an extension of logic. It also explains the difference between mathematics and metamathematics, how consistency and completeness are proven logically, and how Gödel encoded the metamatematics in Principia into mathematical statements. It then explains how Gödel used this encoding to weave a paradox into the system, showing that it is incomplete.

The problem with GEB is that it explains Gödel's proof, but then uses the majority of its pages to advance a philosophical view that human thought is nothing more than computation (one that has been, in my opinion, thoroughly refuted by John Searle). This book sticks to its topic: Gödel's proof and its mathematical implications. It does not stray into speculative philosophy, venturing beyond what Gödel's theorem actually proved.

In short, this book is everything that GEB is not. It is simple, to the point, relatively easy to understand, and sticks to the topic at hand.

The problem with GEB is that it explains Gödel's proof, but then uses the majority of its pages to advance a philosophical view that human thought is nothing more than computation (one that has been, in my opinion, thoroughly refuted by John Searle). This book sticks to its topic: Gödel's proof and its mathematical implications. It does not stray into speculative philosophy, venturing beyond what Gödel's theorem actually proved.

In short, this book is everything that GEB is not. It is simple, to the point, relatively easy to understand, and sticks to the topic at hand.

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ByMidhersandon September 25, 2016

True gem of a book. I have degrees in Mathematics and Physics and I still don't thoroughly understand Godel. This little book helped. Been through it a few times over the years. Highly recommended, especially for those in computer science, logic and number theory.

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ByGregory W. Pappason March 18, 2016

This is an excellent book that makes one of the most difficult subjects to truly understand accessible to someone that has some understanding of number theory. Kurt Godel's Incompleteness Theory is a monument to human thinking on a level equal to Albert Einstein's Theory of Relativity which is only fitting because Einstein was a great admirer and friend of Godel. It is an excellent explanation of the limits that exist of formalized mathematics with respect to the fundamental foundation of number theory.

Ernest Nagel's Godel's Proof could and should be made into a high level college course.

Ernest Nagel's Godel's Proof could and should be made into a high level college course.

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Bytresdfgon January 20, 2012

If Godel proved that no sufficiently complex system, i.e one that is capable of arithmetic, can prove its own consistency or if you assume the system is consistent there will always exist (infinitely many) true statements that cannot be deduced from its axioms, in what system did he prove it in? Is that system consistent? In what sense is the Godel statement true if not by proof? You'll have hundreds of questions popping in your mind every few minutes, and this short book does a very good job of tackling most of them.

Godel numbering is a way to map all the expressions generated by the successive application of axioms back onto numbers, which are themselves instantiated as a "model" of the axioms. The hard part of it is to do this by avoiding the "circular hell". Russell in Principia Mathematica tried hard to avoid the kind of paradoxes like "Set of all elements which do not belong to the set". Godel's proof tries hard to avoid more complicated paradoxes like this :

Let p = "Is a sum of two primes" be a property some numbers might possess. This property can be stated precisely using axioms, and symbols can be mapped to numbers. ( for e.g open a text file, write down the statement and look at its ASCII representation ). The let n(p) be the number corresponding to p. If n(p) satisfies p, then we say n(p) is Richardian, else not. Being Richardian itself is a meta-mathematical property r = "A number which satisfies the property described by its reverse ASCII representation". Note that it is a proper statement represented by the symbols that make up your axioms. Now, you ask if n(r) is Richardian, and the usual problem emerges : n(r) is Richardian iff it is not Richardian. This apparent conundrum, as the authors say, is a hoax. We wanted to represent arithmetical statements as numbers, but switched over to representing meta-mathematical statements as numbers. Godel's proof avoid cheating like this by carefully mirroring all meta-mathematical statements within the arithmetic, and not just conflating the two. Four parts to it.

1. Construct a meta-mathemtical formula G that represents "The formula G is not demonstratable". ( Like Richardian )

2. G is demonstrable if and only if ~G is demonstrable ( Like Richardian)

3. Though G is not demonstrable, G is true in the sense that it asserts a certain arithmetical property which can be exactly defined. ( Unlike Richardian ).

4. Finally, Godel showed that the meta-mathematical statement "if `Arithmetic is consistent' then G follows" is demonstrable. Then he showed that "Arithmetic is consistent" is not demonstrable.

It took me a while to pour over the details, back and forth between pages. I'm still not at the level where I can explain the proof to anyone clearly, but I intend to get there eventually. Iterating is the key.

When I first came across Godel's theorem, I was horrified, dismayed, disillusioned and above all confounded - how can successively applying axioms over and over not fill up the space of all theorems? Now, I'm slowly recuperating. One non-mathematical, intuitive, consoling thought that keeps popping into my mind is : If the axioms to describe arithmetic ( or something of a higher, but finite complexity ) were consistent and complete, then why those axioms? Who ordained them? Why not something else? If it turned out that way, then the question of which is more fundamental : physics or logic would be resolved. I would be shocked if it were possible to decouple the two and rank them - one as more fundamental than the other. I'm very slowly beginning to understand why Godel's discovery was a shock to me.

You see, I'm good at rolling with it while I'm working away, but deep down, I don't believe in Mathematical platonism, or logicism, or formalism or any philosophical ideal that tries to universally quantify.

Kindle Edition - I would advise against the Kindle edition as reading anything with a decent bit of math content isn't a very linear process. Turning pages, referring to footnotes and figures isn't easy on the Kindle.

Godel numbering is a way to map all the expressions generated by the successive application of axioms back onto numbers, which are themselves instantiated as a "model" of the axioms. The hard part of it is to do this by avoiding the "circular hell". Russell in Principia Mathematica tried hard to avoid the kind of paradoxes like "Set of all elements which do not belong to the set". Godel's proof tries hard to avoid more complicated paradoxes like this :

Let p = "Is a sum of two primes" be a property some numbers might possess. This property can be stated precisely using axioms, and symbols can be mapped to numbers. ( for e.g open a text file, write down the statement and look at its ASCII representation ). The let n(p) be the number corresponding to p. If n(p) satisfies p, then we say n(p) is Richardian, else not. Being Richardian itself is a meta-mathematical property r = "A number which satisfies the property described by its reverse ASCII representation". Note that it is a proper statement represented by the symbols that make up your axioms. Now, you ask if n(r) is Richardian, and the usual problem emerges : n(r) is Richardian iff it is not Richardian. This apparent conundrum, as the authors say, is a hoax. We wanted to represent arithmetical statements as numbers, but switched over to representing meta-mathematical statements as numbers. Godel's proof avoid cheating like this by carefully mirroring all meta-mathematical statements within the arithmetic, and not just conflating the two. Four parts to it.

1. Construct a meta-mathemtical formula G that represents "The formula G is not demonstratable". ( Like Richardian )

2. G is demonstrable if and only if ~G is demonstrable ( Like Richardian)

3. Though G is not demonstrable, G is true in the sense that it asserts a certain arithmetical property which can be exactly defined. ( Unlike Richardian ).

4. Finally, Godel showed that the meta-mathematical statement "if `Arithmetic is consistent' then G follows" is demonstrable. Then he showed that "Arithmetic is consistent" is not demonstrable.

It took me a while to pour over the details, back and forth between pages. I'm still not at the level where I can explain the proof to anyone clearly, but I intend to get there eventually. Iterating is the key.

When I first came across Godel's theorem, I was horrified, dismayed, disillusioned and above all confounded - how can successively applying axioms over and over not fill up the space of all theorems? Now, I'm slowly recuperating. One non-mathematical, intuitive, consoling thought that keeps popping into my mind is : If the axioms to describe arithmetic ( or something of a higher, but finite complexity ) were consistent and complete, then why those axioms? Who ordained them? Why not something else? If it turned out that way, then the question of which is more fundamental : physics or logic would be resolved. I would be shocked if it were possible to decouple the two and rank them - one as more fundamental than the other. I'm very slowly beginning to understand why Godel's discovery was a shock to me.

You see, I'm good at rolling with it while I'm working away, but deep down, I don't believe in Mathematical platonism, or logicism, or formalism or any philosophical ideal that tries to universally quantify.

Kindle Edition - I would advise against the Kindle edition as reading anything with a decent bit of math content isn't a very linear process. Turning pages, referring to footnotes and figures isn't easy on the Kindle.

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ByBen Corneron August 15, 2011

Interesting and well-written book. The other reviews here do a great job discussing the content of this book, so I will not linger on that aspect.

You should note that the Kindle Edition had some confusing formatting. For example, here is a screenshot of a page ([...]) where there is a severe lack of formatting for exponents. Should at least read "2^8 x 3^(11^2) x 5^2 x 7^(11^2) x 11^9 x 13^3 x 17^(11^2)", so that it is at least marginally easier to discern where the exponents are. Yes, you ought to be able to figure it out with a moment's thought if you're reading the text that accompanies it, but I thought it was a little disappointing and that it warranted some mention in a review. This has made me cautious of buying any further math or math-related books for my kindle.

You should note that the Kindle Edition had some confusing formatting. For example, here is a screenshot of a page ([...]) where there is a severe lack of formatting for exponents. Should at least read "2^8 x 3^(11^2) x 5^2 x 7^(11^2) x 11^9 x 13^3 x 17^(11^2)", so that it is at least marginally easier to discern where the exponents are. Yes, you ought to be able to figure it out with a moment's thought if you're reading the text that accompanies it, but I thought it was a little disappointing and that it warranted some mention in a review. This has made me cautious of buying any further math or math-related books for my kindle.

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ByEmre Sevincon October 13, 2011

I remember my excitement when I read the first edition of this little gem back in 1999 (actually it was its Turkish translation). Being a young student of mathematics, it was impossible to resist reading a popular and clear account of maybe the most important theorem related to the fundamentals of axiomatic systems. After that came Hofstadter's Gödel, Escher, Bach: An Eternal Golden Braid which introduced more questions related to symbolic logical reasoning, artificial intelligence, cognitive science, and the consequences of Gödel's work in those ares. With that background and ten years after the second edition, it was truly an exciting second reading, a refresher that was both fun and putting lots of things into perspective. Hofstadter's foreword to this edition is a delight to read and ponder upon. On the other hand, I don't think this is a point strong enough to persuade most of the people who own the first edition anyway. But if you don't have the first edition and want a concise and clear explanation of what Gödel's work is all about then this book is definitely for you.

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ByPatrick Walshon May 1, 2013

Very clear and concise. I do have a background in mathematics, but I don't think it'd be hard for anyone who had passed calculus to fully and clearly understand everything involved. They do skip some of the more difficult parts and miss nothing for it.

Overall, a wonderful book on a very difficult theorem. Short enough to allow a reread for clarification, but long enough to get all the important points across.

Written well and there are some amazing results that I never thought possible in metamathematics. The whole Godel numbering system is enough to wow anyone. I had always wondered how it was possible to number mathematical statements. Godel provided an ingenious way to actually do it.

Overall, a wonderful book on a very difficult theorem. Short enough to allow a reread for clarification, but long enough to get all the important points across.

Written well and there are some amazing results that I never thought possible in metamathematics. The whole Godel numbering system is enough to wow anyone. I had always wondered how it was possible to number mathematical statements. Godel provided an ingenious way to actually do it.

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ByTom Grayon August 21, 2016

This book allowed me to understand Godel's proof and its implications. This surprised me and I would imagine even more surprise the professors to attempted to teach me the elements of mathematics. The book is written in an open and accessible style that enabled me to understand the work. I would recommend this book to anyone who wants to understand the often cited but understood by few theorem

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ByKay Mon June 19, 2017

Had to read for class. It was very readable, but it does require a basic understanding of mathematical logic.

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