Customer Reviews

Godel's Proof

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This is a remarkable book. It examines in considerable detail Godel's proof, a mathematical demonstration noted for its difficulty in its novel logical arguments. The chapter topics - the systematic codification of formal logic, an example of a successful absolute proof of consistency, the arithmetization of meta-mathematics - appear almost unapproachable. And yet, Ernest Nagel and James R. Newman have created a delightful exposition of Godel's proof. I actually read this book in one sitting that took me late into the night. I simply didn't want to stop; it is really a good little book.

Godel's proof is not easy to follow, nor easy to grasp the full implications of its conclusions. Many mathematical texts, overviews, and historical summaries avoid directly discussing Godel's proof as these quotes indicate: "Godel's proof is even more abstruse than the beliefs it calls into question." "The details of Godel's proofs in his epoch-making paper are too difficult to follow without considerable mathematical training. "These theorems of Godel are too difficult to consider in their technical details here." Such is the common reference to Kurt Godel's milestone work in logic and mathematics.

In their short book (118 pages) Nagel and Newman present the basic structure of Godel's proof and the core of his conclusions in a way that is intelligible to the persistent layman. This is not an easy book, but it is not overly difficult either. It does require concentration and a willingness to reread some sections, especially the second half.

"Godel's Proof" begins with an explanation of the consistency problem: how can we be assured that an axiomatic system is both complete and consistent? The next chapter reviews relevant mathematical topics, modern formal logic, and places Godel's work in a meaningful historical context. Following chapters explain Hilbert's approach to the consistency problem - the formalization of a deductive system, the meaning of model-based consistency versus absolute consistency, and gives an example of a successful absolute proof of consistency.

The plot now begins to twist and turn. We learn about the Richardian Paradox, an unusual mapping that proves to be logically flawed, but nonetheless provided Godel with a key to mapping meta-mathematics to an axiomatic deductive system. (I forgot to explain meta-mathematics; you will need to read the story.) And then we learn about Godel numbering, a mind boggling way to transform mathematical statements into arithmetic quantities. This novel approach leads to conclusions that shake the foundations of axiomatic logic!

The authors carefully explore and explain Godel's conclusions. For the first time I began to comprehend Godel's fundamental contribution to mathematics and logic. I am almost ready to turn to Godel's original work (in translation), his 1931 paper titled "On Formally Undecidable Propositions of Principia Mathematica and Related Systems". But first, I want to read this little book, this little gem, a few more times.

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!

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.

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.

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

It gives me the same feeling after reading "Brief History of Time". They both explain some very fundamental thing in Science in layman's term. But the difference from "Brief History of Time" is that I can fully understand what the authors are trying to convey.

The footnotes are very helpful in clarifing the terms and concepts used in the main body. I would suggest you not to skip those valuable footnotes.

The whole book is not hard to understand, although you may have trouble reading Section 7: Godel's Proofs. But just go slowly (don't pause in the middle, otherwise you may forget what a particular symbol means) and everything is fine. This Section is the most exciting part of the whole book.

As a Math Grad, this book makes clear to me some concepts that I was not so sure before. One of these corrected concepts is: Godel only ruled out the possibility of getting a proof of consistency within arithmetic. So there is still a hope (though quite unlikely) of finding the proof not representable in arithmetic. See the last section of the book for details.

This is an excellent summary of Godel's main ideas. If you want to understand Godel's proof in less than 100 pages, look no further.

This is an excellent book. No nonsense, no cute metaphors (except one involving supermarkets. But one is excusable!) Douglas Hofstadter's "Gödel, Escher, Bach", though entertaining, tended to rely on obscure (and interminable) form-content interplay to illustrate its points. "Gödel's Proof" sticks to the facts and gets you there in a few hours.

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.)

Early in the second decade of the twentieth century, Bertrand Russell and Alfred Whitehead published their monumental work "Principia Mathematica". In it, they claimed to have laid out the mathematical foundations on top of which the demonstration of all true propositions could be constructed.

However, Kurt Gödel's milestone publication of 1931 exposed fundamental limitations of any axiomatic system of the kind presented in "Principia Mathematica". In essence, he proved that if any such axiomatic system is consistent (i.e., does not contain a contradiction) then there will necessarily exist undecidable propositions (i.e., propositions that can not be demonstrated) that are nevertheless true. The original presentation of Gödel's result is so abstract that it is accessible to only a few specialists within the field of number theory. However, the implications of this result are so far reaching that it has become necessary over the years to make Gödel's ideas accessible to the wider scientific community.

In this book, Nagel and Newman provide an excellent presentation of Gödel's proof. By stripping away some of the rigor of the original paper, they are able to walk the reader through all of Gödel's chain of thought in an easily understandable way. The book starts by paving the way with a few preparatory chapters that introduce the concept of consistency of an axiomatic system, establish the difference between mathematical and meta-mathematical statements, and show how to map every symbol, statement and proof in the axiomatic system on to a subset of the natural numbers. By the time you reach the crucial chapter that contains Gödel's proof itself all ideas are so clear that you'll be able to follow every argument swiftly.

The foreword by Douglas Hofstadter puts the text of this book into the context of twenty-first century thinking and points out some important philosophical consequences of Gödel's proof.