» Godel's Proof

Godel's Proof Reviews

Customer Rating:
Summary: Godel for people such as we (who are familiar with a little Theory of Numbrets)
Comment: I ran into Godel back in about 1955 in "Scientific American". I did not understand that. Now 53 years later, and with some more understanding of the theory of numbers, I find this work to be a magnificent opening into Godel's world. AS WOMDERFUL read!.

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Customer Rating:
Summary: A Simple Presentation of a Complex Topic
Comment: Godel's proof represents a milestone in mathematical and philosophical thought. This book, annotated by the remarkable Douglas Hofstadter, presents Godel's ideas in a language that's readily understandable by the educated layperson. It is clear, concise, and fascinating.

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Customer Rating:
Summary: Godel's incompleteness theorems explained in non-technical language
Comment:
There is no question in my mind that the most misunderstood mathematical theorems of all time are Godel's incompleteness theorems. In essence, they state that no system powerful enough to do basic arithmetic is complete. Meaning that there will always be statements that are true in the system but that can never be proven in that system. These results have been seized upon by people with many different agendas and used to argue conclusions as significant as the existence of God and that human intelligence is not simply the outward manifestation of neurotransmitters flowing from place to place. While this is all somewhat amusing, it is also disquieting, as the theorems cannot be used to conclusively justify such significant claims.
This book is one of the very first books where an attempt is made to explain Godel's theorems to the mathematical laity. In that sense, it is a success; the appropriate background is effectively put forward before the theorem and proof are explained. There is little in the way of formal mathematics and the bulk of the terminology is non-technical. If the people who use Godel's results to justify their extravagant claims were to read this book with an open mind, they would recognize the absurdity of their positions.

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Customer Rating:
Summary: Godel's incompleteness theorem, clearly explained
Comment: Gödel's incompleteness theorem remains one of the most quoted, yet most misunderstood work in mathematics of the last century. Many non mathematicians had used the theorem (without understanding it, of course), to "prove" just about everything (generally it's been used to imply the limits of science, or stuff to that effect). The theorem, though, hardly implies that. This short book, written some 50 years ago, remains probably the best explanation of Godel's work available to the layman. The book starts explaining the background to Godel's theorem, as mathematicians such as Hilbert and Russell sought the axiomatization of mathematics. Godel's work, of course, proved that to be impossible. The book then proceeds to explain the theorem itself, as clearly as it possibly can (though I have to say that, as a non expert, the Gödel numbering scheme seems a like a trick to me, a sleight of hand. Yet, what do I know about this?). Overall, a great book about a much misunderstood work.

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Customer Rating:
Summary: how i understand Godel
Comment: Godel was able to construct a formula from the axioms of Principia Mathematica (PM, and related systems, due to Russell and Whitehead) that is (roughly) "There exists no proof for this formula". This IS the formula itself. now, we need to know if this is true or not. so we try to find a proof for it within PM. if it is possible to find the proof, that means the formula is correct. but the formula says you can't find it. if the proof cannot be found, that means the negation of the formula is correct, but then the formula tells you that you cannot find it, so the formula must be correct, not its negation. in another words, the formula is undecidable within PM. also meaning PM is incomplete. what is amazing now is, of course that is what the formula tells you. so we do know it is true eventhough we can't show it within PM.

note that some people like reviewer Paul Vjecsner, who has also posted his disagreements on other Godel-related books, still confuse mathematics and meta-mathematics. although i have described and mixed-up meta-mathematics meanings to the formula above, Godel's proof was completely mathematicized within PM. either Vjecsner didn't understand the proof or he underestimated the grand aim of PM and thus the significance of Godel's work in taking it apart. Vjecsner argues that Godel's proof is essentially a linguistic paradox that has been unreasonably translated into PM. but the proof doesn't need that. that explanation is only done to give the readers a vague glimpse of the proof in a meta-mathemtical level. the proof only shows that there is a formula which is decidable if its negation is decidable. if you then argue that this is an unacceptable formula, then you are making even bigger claim than Godel, namely that the system is inconsistent! but the problem for you is you can't prove that within PM, thus still showing that PM is incomplete. Godel's proof is only as meaningful as PM. if you poke hole at Godel, you are poking hole at PM, which is exactly what Godel wanted to prove.

the book gives you an outline of how Godel went about constructing that formula with the language of PM, how he made the proof number-theoretical, and many more details. of course reading Godel's original paper would still be nightmarishly difficult even for many mathematicians, so the book gives a very good 'Godel's proof for dummies'. so you want to know what that formula looks like? read this book.

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