Customer Rating: 
Summary: This is a strongly dissenting review
Comment: My challenge would no doubt be viewed with contempt, despite the many justified challenges of the status quo in history. Somehow it is always thought that although profound mistakes were made in the past, the contemporary establishment has things straightened out. My contention is that what is called Goedel's proof is a fundamental embarrassment in the history of logic and mathematics.
There are many particulars in this book that I can criticize which extend beyond Goedel. There is the formalization of deductive systems tracing back to Hilbert. It consists in using symbols without meaning, which are afterward "interpreted" according to subject matter. The like happened most prominently in non-Euclidean geometry, when terms like "straight line" and "plane" were reinterpreted to designate certain curved lines and surfaces, to then demonstrate that some Euclidean theorems do not hold. The problem is that this commits the basic fallacy of equivocation. By redefining words, the statements containing them no longer apply to the original meanings. Likewise, meaningless symbols in formal systems can be defined in conflicting ways. Presently this concerns the use of "Goedel numbers" for such symbols.
"Goedel's proof" is about a sentence which may be phrased as
"THIS STATEMENT IS UNPROVABLE (IN THE SYSTEM)".
The sentence has been recast into logical symbols, and it is these symbols, individually or combined, that are given "Goedel numbers". The argument then is that whatever is determined about the sentence applies to the numbers, viz. to mathematics. The equivocation should be obvious, but alas it isn't. The thinking is explained away by examples like the use of algebra in geometry. But these concern actual mathematical content. The sentence now at issue has no such content, the numbering merely designating the linguistic constituents.
Now to the sentence itself. As indicated there inside parentheses, the sentence is said unprovable in whatever deductive system is used, but Goedel is held to have proved it via "meta-mathematical" means outside the system. This contention I see as an excuse, avoiding contradiction. For, whatever the logic in the proof, it can be made part of the system. To exclude valid logical procedures from deductive systems unjustifiably cripples the systems, and to do so only serves the purpose of claiming an impossibility by not utilizing available tools.
This means that the alleged unprovability in logic, let alone mathematics, of statements otherwise decided true is false, and the statements can thus easily be shown to harbor contradictions and accordingly constitute paradoxes, rather than be part of an assumed consistent system. Regarding the above capitalized sentence, the argument advanced can be put as follows.
If the statement were provable then its negation,
"THE STATEMENT IS PROVABLE",
would also be provable, since that is what the provability would mean, and if this negation were provable then the original statement would also be provable, since the negation says so (p.99 in the book, in different words).
Then it is argued (p.100) that since "if a formula and its...negation can both be derived in some formal calculus [deductive system], then the calculus is not consistent", it follows that if it is consistent, "neither the formula...nor its negation can be demonstrable".
(This is said to make the formula "undecidable", which interestingly supposes the system consistent, in saying, p.101, "that an undecidable formula can be constructed within [it]". Here we see another pitfall of "formal", i.e. meaningless, systems. It is somehow thought that if they comply with grammatical rules, there is no problem. But grammatical sentences can in meaning be contradictory.)
It is further argued (pp.101-2) that one of the above capitalized statements must be true, and since the initial one was found "undecidable", namely to have "no proof inside [the system]", it is the true one, by asserting that unprovability.
It is finally reemphasized that this was not deduced "from the axioms and rules of a formal system, but by a meta-mathematical argument."
But those axioms and rules, unlike contended, easily accommodate the logic, which is quite common. That neither of the above capitalized sentences is provable because it implies the other follows by the "law of contradiction"; that one of them must presumably be true follows by the "law of excluded middle"; and that it is the initial sentence which is true, since just found unprovable, can be said to follow by the "law of identity". Thus the unprovable sentence is contradictorily proved via the well-known "laws of thought".
As indicated, the sentence is correspondingly internally contradictory and can be added to other paradoxes, rather than be proclaimed true and unprovable.
Customer Rating:
Summary: Interesting introduction and some very good hints to a conclusion
Comment: I have been a big fan of the issue of formal mathematics and the theory of computation but I always missed a full grasping of the goedel theorem. The book presents the line along which Goedel moved: mostly formal systems and the most interesting issue of "calculating grammar" by powers of primes: an outstanding example, to me, was how using that instrument one could know whether a sentence was introduced by a "not" or not (in the case simply by checking if the figure expressing the formula was even or odd).
In that way the full system of mathematics turned into a sort of a computer program that of course could not calculate every function.
For that matter I came to wonder why the demonstration of goedel theorem could not be carried out simply by showing a formal system has the same power as a universal Turing machine and thus transferring to it the (much easier) results obtained on that issue - like for example to problem of the stop or the one of finding any semantic information in a program without actually executing it.
Customer Rating:
Summary: Perfect for the Technical -and- the Non-Technical Reader
Comment: As a graduate student in mathematics and physics, logic and rigorous deduction play an every day role in my work. But nowhere in my education were the foundations of logic emphasized, and Godel Incompleteness has never been mentioned in a classroom. I was vaguely aware of the names/ideas involved, and I wanted to learn something about them.
I didn't want something so watered down that it would leave me feeling like I knew nothing and just wasted my time. But, after trying to read Godel's original papers, I realized I couldn't just jump directly into the technical literature, despite my background.
I fortunately ended up picking Nagel & Newman's text. The clarity of the writing is utter perfection. There is no background assumed, and even a liberal arts major can understand the entire book. Fortunately, Godel's arguments only require simple arithmetic, so you still come away with an appreciation of the brilliance of his argument, and insight into the methods of his proofs.
This book is the perfect stepping stone for further study into logic & incompleteness. I should also note that Hofstadter's footnotes are a significant addition, so this 2001 revised edition is even more valuable than the original classic. (In Douglas Hofstadter's monumental Pulitzer-winning 1979 book - "Godel, Escher, Bach" - he credits Nagel & Newman's book as being his inspiration.)
Even though it was written in 1958, it's hard to imagine anything being subsequently published that could so eloquently do what Nagel & Newman's text does in a mere 100 pages.
Customer Rating:
Summary: Great Description of Difficult Work - An Excellent Introduction
Comment: I had read "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" By the mathermatician himself and then found Ernest Nagel's "Godel's Proof" nearly by accident. The titles of the work are examples of the main diffeernce of the two: the latter is far more simple and comprehensible. Diving right into Goedels work with a some decent understanding of mathematics and a thourough reading of "Principia Mathematica" by Russel and Whitehead, I thought I would be able to handle it. I was able to comprehend Goedel but found it gave me a headache to read more than a few pages at a time. Getting through after far too many hours and little true understanding. It seemed that while I could grasp the concepts I wasnt so clear on the subtlties of Goedel's theorem. I was more than happy to read Nagel's Work which is very approachable and exemplifies the important points that the average person might breeze through in Goedel's work. This being said the work of Nagel should be considered an introduction to Goedel's work and both have their place as excellent works.
I would recommend that everyone who is interested in the philosophical and mathematical implications of the incompleteness theorem read this work and keep it on hand as they attempt Goedel. I find that people seem to get the basic idea of incompleteness but overextend or misunderstand its reach in life and in meaning. The theorem itself is among the most interesting mathematics and it is a philosophically profound idea that people at large dont grasp since the system of mathematics appears to work well in nearly all situations. This book will be enjoyable and easy to understand even if you dont have a degree in mathematics so long as you tkae it slow but understnading of the Principia and mathematical philosophy is key to getting the most out of this.
Ted Murena
Customer Rating:
Summary: Thoroughness in Explaining Background and Context as well as Gödel's Proof Itself
Comment: I redid my review (now July 2006) after your 50:50 votes on helpfulness. I think you needed more content to the review and less ebullience. So here it is... In the interim, I have read other treatments of Gödel's proof (including the Dover book of Gödel's article itself also with an introduction, Beyond Numeracy, The Advent of the Algorithm [ref below], and several others). What stands out in THIS book, though, is the extreme thoroughness of explaining to you the context in which Gödel was working at the time. This book is unique in its dedication to getting you to a concrete understanding of -- and appreciation for -- the background and context. In fact making sure you get the context appreciation takes up about 2/3rd's of the book! Of course the book is thorough on the Proof itself too. Is that part easy? No, it's still not. But you won't be left at all vague on what the proof is like. The only other book that is as good on the CONTEXT of Gödel's proof is The Advent of the Algorithm. The Advent of the Algorithm is also excellent on how others took, and "ran-with", Gödel's results. As for which edition of this book (Gödel's Proof) to get, the new addition has Hofstaddter's introduction. That intro adds value for sentimentality (if you should so find his story about his reading the book and his subsequent friendship with Nagel) and Hofstadter's own ebullience, but the book is virtually identical otherwise with its 1959 edition. It would be perfectly good -- you'll miss nothing -- if you bought a cheap 1959 edition. For a good complimentary book, get also The Advent of the Algorithm by David Berlinski (2000) ISBN 0 15 100338 6 or ISBN 0 15 601391 6 (pbk). You can read my review on that book too if you like.
Summary: This is a strongly dissenting review
Comment: My challenge would no doubt be viewed with contempt, despite the many justified challenges of the status quo in history. Somehow it is always thought that although profound mistakes were made in the past, the contemporary establishment has things straightened out. My contention is that what is called Goedel's proof is a fundamental embarrassment in the history of logic and mathematics.
There are many particulars in this book that I can criticize which extend beyond Goedel. There is the formalization of deductive systems tracing back to Hilbert. It consists in using symbols without meaning, which are afterward "interpreted" according to subject matter. The like happened most prominently in non-Euclidean geometry, when terms like "straight line" and "plane" were reinterpreted to designate certain curved lines and surfaces, to then demonstrate that some Euclidean theorems do not hold. The problem is that this commits the basic fallacy of equivocation. By redefining words, the statements containing them no longer apply to the original meanings. Likewise, meaningless symbols in formal systems can be defined in conflicting ways. Presently this concerns the use of "Goedel numbers" for such symbols.
"Goedel's proof" is about a sentence which may be phrased as
"THIS STATEMENT IS UNPROVABLE (IN THE SYSTEM)".
The sentence has been recast into logical symbols, and it is these symbols, individually or combined, that are given "Goedel numbers". The argument then is that whatever is determined about the sentence applies to the numbers, viz. to mathematics. The equivocation should be obvious, but alas it isn't. The thinking is explained away by examples like the use of algebra in geometry. But these concern actual mathematical content. The sentence now at issue has no such content, the numbering merely designating the linguistic constituents.
Now to the sentence itself. As indicated there inside parentheses, the sentence is said unprovable in whatever deductive system is used, but Goedel is held to have proved it via "meta-mathematical" means outside the system. This contention I see as an excuse, avoiding contradiction. For, whatever the logic in the proof, it can be made part of the system. To exclude valid logical procedures from deductive systems unjustifiably cripples the systems, and to do so only serves the purpose of claiming an impossibility by not utilizing available tools.
This means that the alleged unprovability in logic, let alone mathematics, of statements otherwise decided true is false, and the statements can thus easily be shown to harbor contradictions and accordingly constitute paradoxes, rather than be part of an assumed consistent system. Regarding the above capitalized sentence, the argument advanced can be put as follows.
If the statement were provable then its negation,
"THE STATEMENT IS PROVABLE",
would also be provable, since that is what the provability would mean, and if this negation were provable then the original statement would also be provable, since the negation says so (p.99 in the book, in different words).
Then it is argued (p.100) that since "if a formula and its...negation can both be derived in some formal calculus [deductive system], then the calculus is not consistent", it follows that if it is consistent, "neither the formula...nor its negation can be demonstrable".
(This is said to make the formula "undecidable", which interestingly supposes the system consistent, in saying, p.101, "that an undecidable formula can be constructed within [it]". Here we see another pitfall of "formal", i.e. meaningless, systems. It is somehow thought that if they comply with grammatical rules, there is no problem. But grammatical sentences can in meaning be contradictory.)
It is further argued (pp.101-2) that one of the above capitalized statements must be true, and since the initial one was found "undecidable", namely to have "no proof inside [the system]", it is the true one, by asserting that unprovability.
It is finally reemphasized that this was not deduced "from the axioms and rules of a formal system, but by a meta-mathematical argument."
But those axioms and rules, unlike contended, easily accommodate the logic, which is quite common. That neither of the above capitalized sentences is provable because it implies the other follows by the "law of contradiction"; that one of them must presumably be true follows by the "law of excluded middle"; and that it is the initial sentence which is true, since just found unprovable, can be said to follow by the "law of identity". Thus the unprovable sentence is contradictorily proved via the well-known "laws of thought".
As indicated, the sentence is correspondingly internally contradictory and can be added to other paradoxes, rather than be proclaimed true and unprovable.
Customer Rating:
Summary: Interesting introduction and some very good hints to a conclusion
Comment: I have been a big fan of the issue of formal mathematics and the theory of computation but I always missed a full grasping of the goedel theorem. The book presents the line along which Goedel moved: mostly formal systems and the most interesting issue of "calculating grammar" by powers of primes: an outstanding example, to me, was how using that instrument one could know whether a sentence was introduced by a "not" or not (in the case simply by checking if the figure expressing the formula was even or odd).
In that way the full system of mathematics turned into a sort of a computer program that of course could not calculate every function.
For that matter I came to wonder why the demonstration of goedel theorem could not be carried out simply by showing a formal system has the same power as a universal Turing machine and thus transferring to it the (much easier) results obtained on that issue - like for example to problem of the stop or the one of finding any semantic information in a program without actually executing it.
Customer Rating:
Summary: Perfect for the Technical -and- the Non-Technical Reader
Comment: As a graduate student in mathematics and physics, logic and rigorous deduction play an every day role in my work. But nowhere in my education were the foundations of logic emphasized, and Godel Incompleteness has never been mentioned in a classroom. I was vaguely aware of the names/ideas involved, and I wanted to learn something about them.
I didn't want something so watered down that it would leave me feeling like I knew nothing and just wasted my time. But, after trying to read Godel's original papers, I realized I couldn't just jump directly into the technical literature, despite my background.
I fortunately ended up picking Nagel & Newman's text. The clarity of the writing is utter perfection. There is no background assumed, and even a liberal arts major can understand the entire book. Fortunately, Godel's arguments only require simple arithmetic, so you still come away with an appreciation of the brilliance of his argument, and insight into the methods of his proofs.
This book is the perfect stepping stone for further study into logic & incompleteness. I should also note that Hofstadter's footnotes are a significant addition, so this 2001 revised edition is even more valuable than the original classic. (In Douglas Hofstadter's monumental Pulitzer-winning 1979 book - "Godel, Escher, Bach" - he credits Nagel & Newman's book as being his inspiration.)
Even though it was written in 1958, it's hard to imagine anything being subsequently published that could so eloquently do what Nagel & Newman's text does in a mere 100 pages.
Customer Rating:
Summary: Great Description of Difficult Work - An Excellent Introduction
Comment: I had read "On Formally Undecidable Propositions of Principia Mathematica and Related Systems" By the mathermatician himself and then found Ernest Nagel's "Godel's Proof" nearly by accident. The titles of the work are examples of the main diffeernce of the two: the latter is far more simple and comprehensible. Diving right into Goedels work with a some decent understanding of mathematics and a thourough reading of "Principia Mathematica" by Russel and Whitehead, I thought I would be able to handle it. I was able to comprehend Goedel but found it gave me a headache to read more than a few pages at a time. Getting through after far too many hours and little true understanding. It seemed that while I could grasp the concepts I wasnt so clear on the subtlties of Goedel's theorem. I was more than happy to read Nagel's Work which is very approachable and exemplifies the important points that the average person might breeze through in Goedel's work. This being said the work of Nagel should be considered an introduction to Goedel's work and both have their place as excellent works.
I would recommend that everyone who is interested in the philosophical and mathematical implications of the incompleteness theorem read this work and keep it on hand as they attempt Goedel. I find that people seem to get the basic idea of incompleteness but overextend or misunderstand its reach in life and in meaning. The theorem itself is among the most interesting mathematics and it is a philosophically profound idea that people at large dont grasp since the system of mathematics appears to work well in nearly all situations. This book will be enjoyable and easy to understand even if you dont have a degree in mathematics so long as you tkae it slow but understnading of the Principia and mathematical philosophy is key to getting the most out of this.
Ted Murena
Customer Rating:
Summary: Thoroughness in Explaining Background and Context as well as Gödel's Proof Itself
Comment: I redid my review (now July 2006) after your 50:50 votes on helpfulness. I think you needed more content to the review and less ebullience. So here it is... In the interim, I have read other treatments of Gödel's proof (including the Dover book of Gödel's article itself also with an introduction, Beyond Numeracy, The Advent of the Algorithm [ref below], and several others). What stands out in THIS book, though, is the extreme thoroughness of explaining to you the context in which Gödel was working at the time. This book is unique in its dedication to getting you to a concrete understanding of -- and appreciation for -- the background and context. In fact making sure you get the context appreciation takes up about 2/3rd's of the book! Of course the book is thorough on the Proof itself too. Is that part easy? No, it's still not. But you won't be left at all vague on what the proof is like. The only other book that is as good on the CONTEXT of Gödel's proof is The Advent of the Algorithm. The Advent of the Algorithm is also excellent on how others took, and "ran-with", Gödel's results. As for which edition of this book (Gödel's Proof) to get, the new addition has Hofstaddter's introduction. That intro adds value for sentimentality (if you should so find his story about his reading the book and his subsequent friendship with Nagel) and Hofstadter's own ebullience, but the book is virtually identical otherwise with its 1959 edition. It would be perfectly good -- you'll miss nothing -- if you bought a cheap 1959 edition. For a good complimentary book, get also The Advent of the Algorithm by David Berlinski (2000) ISBN 0 15 100338 6 or ISBN 0 15 601391 6 (pbk). You can read my review on that book too if you like.