Customer Reviews

Godel's Proof

5 star:		(10)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(2)

 

 

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!

I'm an aspiring mathematician/physicist with a strong interest in philosophy. Godel's theorems have always interested me, but I just haven't yet had a chance to study the mathematical logic necessary to understand them. While at the library to pick up a book to begin studying this very subject, I happened to see this book entitled "Godel's Proof". As it was short and purported to give a quick and somewhat general explanation of his work, I decided there couldn't be anything to lose in picking up this book.

I'm happy to say that I read half of it yesterday and am extremely excited to finish off the second half today. As many reviewers have said, this book isn't the rigorous exposition that a logician or mathematician may desire. However, as anyone who studies difficult subjects deeply knows, it is extremely useful to be given an understandable and simple outline of what you are about to embark on. This book (so far) is exactly that. It isn't so rigorous that you get bogged down in details of specific proofs, but it isn't so "dumbed down" that the explanation is devoid of any real meaning.

I recommend this book to anyone in my situation. Someone who would like a simple and understandable (but still valuable) overview of the subject as a precursor to deeper study.

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.)

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.

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!

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.

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.

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.

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.

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.