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Godel's Proof [Hardcover]

Ernest Nagel , James R. Newman , Douglas R. Hofstadter
4.5 out of 5 stars  See all reviews (62 customer reviews)

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Book Description

October 1, 2001 0814758169 978-0814758168 Revised

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 discovery. It offers every educated person with a taste for logic and philosophy the chance to understand a previously difficult and inaccessible subject.

New York University Press is proud to publish this special 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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Editorial Reviews

Amazon.com Review

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."


"An excellent nontechnical account of the substance of Gödel's celebrated paper."

-American Mathematical Society,

Product Details

  • Hardcover: 125 pages
  • Publisher: NYU Press; Revised edition (October 1, 2001)
  • Language: English
  • ISBN-10: 0814758169
  • ISBN-13: 978-0814758168
  • Product Dimensions: 8.2 x 5.2 x 0.6 inches
  • Shipping Weight: 9.6 ounces (View shipping rates and policies)
  • Average Customer Review: 4.5 out of 5 stars  See all reviews (62 customer reviews)
  • Amazon Best Sellers Rank: #577,422 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
156 of 161 people found the following review helpful
5.0 out of 5 stars An Abstruse Mathematical Proof Made Fascinating December 24, 2001
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?
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68 of 71 people found the following review helpful
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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55 of 57 people found the following review helpful
5.0 out of 5 stars An excellent guide to Gödel June 26, 1999
By A Customer
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.
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37 of 38 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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25 of 25 people found the following review helpful
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
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Most Recent Customer Reviews
5.0 out of 5 stars An Excellent Treatment of a Difficult Subject
This edition is an update to the original one which I read for a college course in Symbolic Logic. In the new edition, there are corrections to some key arguments.
Published 5 months ago by David Arp
5.0 out of 5 stars Read This Instead of Gödel, Escher, Bach
This book is simple, short, and to the point. It explains the project of Principia Mathematica, the idea of building mathematics as an extension of logic. Read more
Published 6 months ago by Drew
4.0 out of 5 stars Easy read, short on details
You'll definitely understand Godel's proof by the time you finish reading this short book but you'll be left hungry for details... Read more
Published 6 months ago by Aniketos
5.0 out of 5 stars Great little book
This classic shortmbook offers a great overview of a fascinating result from mathematics. While accessible to anyone who has had little math beyond high school, it still conveys... Read more
Published 8 months ago by CraigM
5.0 out of 5 stars Very readable and inspiring explanation of an important...
This is a concise and understandable elucidation of a complex but far reaching proof. Great companion to Gödel, Escher Bach.
Published 11 months ago by David M. Blocher
5.0 out of 5 stars Stunningly accessible introduction
This is the place to start for those looking to get a handle on Gödel's incompleteness theorem. Read more
Published 12 months ago by whiteelephant
5.0 out of 5 stars Great presentation
Great presentation of Godel's proof. It has been some time since I have studied math proofs and found this presentation helpful.
Published 12 months ago by George M. Feder
5.0 out of 5 stars Godel's Proof - a road to deeper understandingof what knowledge is
The revolutionary Incompleteness Theorems Kurt Godel prove in 1931 are mathematical theorems, that have not only turned the evovement of math upside down but also created immense... Read more
Published 13 months ago by Yaakov Lichter
5.0 out of 5 stars Awesome
Not only is the author able to convey the main ideas and implications of Goedel's Theorem, but he amazingly manages to actually walk the reader through the proof step by step... Read more
Published 13 months ago by Jeremy Schmitt
5.0 out of 5 stars Concise
Very clear and concise. I do have a background in mathematics, but I don't think it'd be hard for anyone who had passed calculus to fully and clearly understand everything... Read more
Published 14 months ago by Patrick Walsh
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