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Title: Godel's Proof by Ernest Nagel, James Roy Newman, Douglas R. Hofstadter ISBN: 0-8147-5816-9 Publisher: New York University Press Pub. Date: February, 2002 Format: Hardcover Volumes: 1 List Price(USD): $17.95 |
Average Customer Rating: 4.4 (20 reviews)
Rating: 5
Summary: Wish I'd read it first ...
Comment: 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!
Rating: 5
Summary: A Must Read for Math and Philosophy Students
Comment: 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.
Rating: 5
Summary: A great summary
Comment: 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.
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Title: On Formally Undecidable Propositions of Principia Mathematica and Related Systems by Kurt Gödel ISBN: 0486669807 Publisher: Dover Pubns Pub. Date: April, 1992 List Price(USD): $6.95 |
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Title: Gödel, Escher, Bach: An Eternal Golden Braid by Douglas R. Hofstadter ISBN: 0465026567 Publisher: HarperCollins Pub. Date: January, 1999 List Price(USD): $21.00 |
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Title: The Principles of Mathematics by Bertrand Arthur Russell ISBN: 0393314049 Publisher: W.W. Norton & Company Pub. Date: February, 1996 List Price(USD): $17.95 |
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Title: Computability and Unsolvability by Martin Davis ISBN: 0486614719 Publisher: Dover Pubns Pub. Date: December, 1985 List Price(USD): $14.95 |
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Title: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire ISBN: 0309085497 Publisher: Joseph Henry Press Pub. Date: 23 April, 2003 List Price(USD): $27.95 |
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