Top positive review
6 people found this helpful
5.0 out of 5 starsAs promised, a complete though not thorough explanation of Godel's incompleteness theorem
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.
Top critical review
3.0 out of 5 starsFastidious author make reading difficult
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).
Search
Sort by
Top rated
There was a problem filtering reviews right now. Please try again later.
5.0 out of 5 starsAs promised, a complete though not thorough explanation of Godel's incompleteness theorem
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.
6 people found this helpful
Helpful
Not Helpful
ByRoberto Rigolin F Lopeson November 28, 2017
We are in 1931, some clever people like Hilbert, Russell and Whitehead have been trying to put mathematics on solid foundations. But a young guy called Gödel just constructed a proof stating that all consistent systems are incomplete. Then if mathematics has no contradictions (consistent), it cannot be composed by theorems (incomplete). Therefore, it is impossible to validate mathematics using mathematics. Damn, this kid just shocked the whole scientific community. How on earth someone proof that ALL consistent systems within the whole universe of possible systems are incomplete? This book will give you some clues. If you want to see and hear it, read GEB by Douglas Hofstadter.
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.
ByAmazon Customeron April 21, 2018
Personally, I think the author did good job as he has tried to explain this very complex topic in relatively simple words. However, this book is intented for somebody already acquainted with mathematics, mainly with basic of mathematical (or formal) logic. The author describes mainly the topic itself, only sometimes he deals with historical backdrop. So, I would not recommend this book to somebody interested in history of science.
The book is divided to two main parts. In the first one (chapters 1 to 6) the author brings introduction to terms like a "consistency" and "completeness" in logic and he desribes methods how to prove that some formal system is consistent or complete. The author is also talking about some logical paradoxes. Then some necessary information about mapping in maths is provided. The second part of the book (namely chapter 7) contains Godel's proof itself. Since the proof is not very simple, the author fisrtly introduces some other auxiliary theorems. Finnaly, the proof is coined and explained in "natural" language. I highly appreciated many footnotes sheding more light on, at first glance, not very clear terms and statements.
To sum up, the book is worth reading and I think that everybody either using advanced maths in day-to-day working or dealing with computer sciences should read this book.
The book is divided to two main parts. In the first one (chapters 1 to 6) the author brings introduction to terms like a "consistency" and "completeness" in logic and he desribes methods how to prove that some formal system is consistent or complete. The author is also talking about some logical paradoxes. Then some necessary information about mapping in maths is provided. The second part of the book (namely chapter 7) contains Godel's proof itself. Since the proof is not very simple, the author fisrtly introduces some other auxiliary theorems. Finnaly, the proof is coined and explained in "natural" language. I highly appreciated many footnotes sheding more light on, at first glance, not very clear terms and statements.
To sum up, the book is worth reading and I think that everybody either using advanced maths in day-to-day working or dealing with computer sciences should read this book.
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.
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
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.
ByAmazon Customeron June 22, 2011
Godel's Proof is a well written and understandable outline of the famous incompleteness theorem. The authors' intents were to make the proof accessible to an audience that may not be familiar with formal logic, or any of the other required foundation of the proof. It does not go over the proof in absolute detail, but it helps the reader understand where the proof came from, defines enough of a vocabulary to get through the basics, and is through enough to convince the reader that the details can be worked out given enough time and effort.
Readers that have already had an extensive education in the area of formal logic will not get very much out of this book. It is written for a very different audience. On the other hand, if you are curious about Godel's theorems and don't have this background, check this book out.
Readers that have already had an extensive education in the area of formal logic will not get very much out of this book. It is written for a very different audience. On the other hand, if you are curious about Godel's theorems and don't have this background, check this book out.
ByCraigMon November 2, 2013
This classic shortmbook offers a great overview of a fascinating result from mathematics. While accessible to anyone who has had little math beyond high school, it still conveys the magic of this proof about the limits of any reasonably complex description of numbers which still corresponds to our intuitive notions of what is right. If you like this humbling result about math, do a little digging for Arrow's Impossibility Theorem and see what a little math can do to upend your notions about reasonable ways for a society to make fair decisions.....
ByKay Mon June 19, 2017
Had to read for class. It was very readable, but it does require a basic understanding of mathematical logic.
There's a problem loading this menu right now.
Get fast, free shipping with Amazon Prime
Prime members enjoy FREE Two-Day Shipping and exclusive access to music, movies, TV shows, original audio series, and Kindle books.
|
Your recently viewed items and featured recommendations
After viewing product detail pages, look here to find an easy way to navigate back to pages you are interested in.
Back to top
Get to Know Us
Make Money with Us
Amazon Payment Products
- Conditions of Use
- Privacy Notice
- Interest-Based Ads
- © 1996-2018, Amazon.com, Inc. or its affiliates
