Godel's Proof

Rating:
Summary: A Must Read for Math and Philosophy Students
Review: 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.

Rating:
Summary: Excellent summary.
Review: Gödel's brilliant incompleteness theorem is astounding. He proves that every system, even that of the arithmetic integers, is inconsistent, and, essentially, he shows us that human intellect cannot be fully formalized. A tour de force.

This book is a summary of Gödels article, together with a comment. The mathematics are certainly not easy to understand and ask more than a few hours to follow the reasoning. But finally one is fascinated by the dazzling brightness of this mathematical construction. An important read.

Rating:
Summary: Wish I'd read it first ...
Review: 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!

Rating:
Summary: Lucid & satisfying: Godel's Proof and modern logic
Review: 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 [e-mail].)

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Summary: It's like "Brief History of Time" in Mathematics
Review: 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.

Rating:
Summary: Exceptionally clear
Review: The beauty of this book is that Godel's ideas and proof is explained with a minimum of symbolic strings. It's exceptionally lucid and the best introduction to Godel I've encountered.

The introduction by Hofstadter is poignant and informative.

Rating:
Summary: Don't be intimidated by the subject matter.
Review: 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.

Rating:
Summary: A great summary
Review: This is a fantastic book that makes the important discoveries of Godel accessible to all interested readers. It often serves as a foyer to all the other literature in this field, including Godel's original, Cantor, Frege and others. A fantastic and clear little book! A gem.

Rating:
Summary: An Abstruse Mathematical Proof Made Fascinating
Review: 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.

Rating:
Summary: Good attempt to explain the proof
Review: 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!