Godel's Proof Hardcover – Oct 1 2001
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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
"An excellent nontechnical account of the substance of Godel's celebrated paper." -American Mathematical Society "A little masterpiece of exegesis." -NatureSee all Product Description
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Top Customer Reviews
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.Read more ›
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 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.
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.
Most recent customer reviews
In 100 lucid and highly readable pages, presents the most important ideas of modern logic: axiomatisation (Euclid), formalization (Hilbert), metamathematical argumentation,... Read morePublished on May 18 2004 by Stavros Macrakis
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 morePublished on Jan. 3 2004 by C. Goss
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. Read morePublished on May 1 2003 by Ken Braithwaite
This is a fantastic book that makes the important discoveries of Godel accessible to all interested readers. Read morePublished on Jan. 13 2003 by rationalist
The beauty of this book is that Godel's ideas and proof is explained with a minimum of symbolic strings. Read morePublished on Sept. 16 2002
The beauty of this book is that Godel's ideas and proof is explained with a minimum of symbolic strings. Read morePublished on Sept. 16 2002 by Mark Twain
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 morePublished on Sept. 5 2002 by Luc REYNAERT
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 morePublished on May 4 2002 by murino man