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Godel's Proof Hardcover – Oct 1 2001

4.3 out of 5 stars 13 customer reviews

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Product Details

  • Hardcover: 129 pages
  • Publisher: New York Univ Pr; Revised ed. edition (Oct. 1 2001)
  • Language: English
  • ISBN-10: 0814758169
  • ISBN-13: 978-0814758168
  • Product Dimensions: 14 x 1.3 x 21.6 cm
  • Shipping Weight: 272 g
  • Average Customer Review: 4.3 out of 5 stars 13 customer reviews
  • Amazon Bestsellers Rank: #688,944 in Books (See Top 100 in Books)
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Product Description

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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."


"An excellent nontechnical account of the substance of G del's celebrated paper.")-(American Mathematical Society), ()


"A little masterpiece of exegesis." )-("Nature"), ()

"A little masterpiece of exegesis." -"Nature",

"Because whistleblowers leaked the Abu Ghraib photos and some of the torture memos, the torture and abuse committed by the United States entered the national discourse. This book is the result of those efforts and this critical work by leading scholars and journalists who courageously provide a roadmap for holding Bush officials accountable for their war crimes."
-Daniel Ellsberg, author of "Secrets: A Memoir of Vietnam and the Pentagon Papers"

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"A magnificent, though deeply disturbing collection of essays on torture, considering its history, its use since September 11, and the obstacles to holding those responsible accountable. This is the best collection of essays on the topic and it leaves no doubt that the nation has not yet come to grips with the inhumanity perpetrated under the guise of national security."
-Erwin Chemerinsky, Dean and Distinguished Professor of Law, University of California, Irvine, School of Law

"An excellent addition to the cannon of work relating to the post-9/11 embrace of torture by the Bush Administration as well as the subsequent erosion of constitutional and international legal principles."-Adam L. Kress, "Law and Politics Book Review"

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-Janis L. Karpinski, former Brigadier General, U.S. Army, and author of "One Woman's Army: The Commanding General of Abu Ghraib Tells Her Story"

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

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

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

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Top Customer Reviews

Format: Hardcover
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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Format: Hardcover
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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Format: Paperback
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.
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Format: Hardcover
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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