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Godel's Proof [Hardcover]

Ernest Nagel , James R. Newman , Douglas R. Hofstadter
4.3 out of 5 stars  See all reviews (13 customer reviews)
Price: CDN$ 28.15 & FREE Shipping. Details
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Book Description

February 2002
"A little masterpiece of exegesis." -Nature

"An excellent nontechnical account of the substance of Gödel's celebrated paper." -American Mathematical Society

In 1931 Kurt Gödel published his fundamental paper, "On Formally Undecidable Propositions of -Principia Mathematica- and Related Systems." This revolutionary paper challenged certain basic assumptions underlying much research in mathematics and logic. Gödel received public recognition of his work in 1951 when he was awarded the first Albert Einstein Award for achievement in the natural sciences-perhaps the highest award of its kind in the United States. The award committee described his work in mathematical logic as "one of the greatest contributions to the sciences in recent times."

However, few mathematicians of the time were equipped to understand the young scholar's complex proof. Ernest Nagel and James Newman provide a readable and accessible explanation to both scholars and non-specialists of the main ideas and broad implications of Gödel's proof. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.

Marking the 70th anniversary of the original publication of Gödel's Proof, New York University Press is proud to publish this special anniversary edition of one of its bestselling books. With a new introduction by Douglas R. Hofstadter, this book will appeal students, scholars, and professionals in the fields of mathematics, computer science, logic and philosophy, and science.

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Gödel's incompleteness theorem--which showed that any robust mathematical system contains statements that are true yet unprovable within the system--is an anomaly in 20th-century mathematics. Its conclusions are as strange as they are profound, but, unlike other recent theorems of comparable importance, grasping the main steps of the proof requires little more than high school algebra and a bit of patience. Ernest Nagel and James Newman's original text was one of the first (and best) to bring Gödel's ideas to a mass audience. With brevity and clarity, the volume described the historical context that made Gödel's theorem so paradigm-shattering. Where the first edition fell down, however, was in the guts of the proof itself; the brevity that served so well in defining the problem made their rendering of Gödel's solution so dense as to be nearly indigestible.

This reissuance of Nagel and Newman's classic has been vastly improved by the deft editing of Douglas Hofstadter, a protégé of Nagel's and himself a popularizer of Gödel's work. In the second edition, Hofstadter reworks significant sections of the book, clarifying and correcting here, adding necessary detail there. In the few instances in which his writing diverges from the spirit of the original, it is to emphasize the interplay between formal mathematical deduction and meta-mathematical reasoning--a subject explored in greater depth in Hofstadter's other delightful writings. --Clark Williams-Derry


"A little masterpiece of exegesis. Nature An excellent non-technical account of the substance of Gdels celebrated paper. Bulletin of the American Mathematical Society

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4.3 out of 5 stars
4.3 out of 5 stars
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5 of 5 people found the following review helpful
5.0 out of 5 stars A Must Read for Math and Philosophy Students March 25 2003
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.
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2 of 2 people found the following review helpful
4.0 out of 5 stars Good attempt to explain the proof May 23 2004
By A Customer
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!
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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.
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5.0 out of 5 stars Wish I'd read it first ... May 1 2003
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!
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Most recent customer reviews
5.0 out of 5 stars Half way through, great read
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... Read more
Published on July 18 2011 by dp
5.0 out of 5 stars Lucid & satisfying: Godel's Proof and modern logic
In 100 lucid and highly readable pages, presents the most important ideas of modern logic: axiomatisation (Euclid), formalization (Hilbert), metamathematical argumentation,... Read more
Published on May 18 2004 by Amazon Customer
5.0 out of 5 stars Don't be intimidated by the subject matter.
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... Read more
Published on Jan. 3 2004 by C. Goss
5.0 out of 5 stars A great summary
This is a fantastic book that makes the important discoveries of Godel accessible to all interested readers. Read more
Published on Jan. 13 2003 by rationalist
5.0 out of 5 stars Exceptionally clear
The beauty of this book is that Godel's ideas and proof is explained with a minimum of symbolic strings. Read more
Published on Sept. 16 2002
5.0 out of 5 stars Exceptionally clear
The beauty of this book is that Godel's ideas and proof is explained with a minimum of symbolic strings. Read more
Published on Sept. 16 2002 by Mark Twain
5.0 out of 5 stars Excellent summary.
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... Read more
Published on Sept. 5 2002 by Luc REYNAERT
1.0 out of 5 stars He Dumbs It Down Way Too Much
As the reviewer below, I too am a medical student/mathematican. I consider myself the foremost amature number theorist since Fermat and can read and translate ANY mathematical or... Read more
Published on May 4 2002 by murino man
1.0 out of 5 stars Only for a novice
I read Godel's original paper over my winter break (2001) from my second year of medical school at UCSF. Read more
Published on Jan. 25 2002
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