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Reading through the reviews of self-proclaimed math geniuses (see some of the below unhelpful reviews for examples) is hardly edifying, so I feel compelled to lend a hand. Here are a few comments about this publication:
First, the introduction does a poor job in explicating the theory. I suppose it gives you the basic idea, but this is hardly the first account of the theory one should read. Brathwaite does not connect all of the dots, and it will take a long time to figure out how the proof works from his intro, if you can do it all. (And that's not a challenge or insult; it simply isn't that well written.)
Second, forget about wading through Godel's proof on your own. The reviewer who claimed to do so with two years of algebra and a really good dictionary is simply lying. You do not wade through difficult theorems in mathematical logic without the appropriate tools. And the appropriate tools include having done similar but simpler proofs on your own and having a solid background in mathematical logic. Without this background, it doesn't matter whether you have the ability to be a mathematics professor at Princeton or place top five in the Putnam - you simply will not understand the proof in a rigorous manner. By all means, take a look at it to get a general feel for what's going on, but if you want a semi-technical account read Smullyan's "Godel's Incompleteness Theorems."
Third, as one reviewer pointed out, there are multiple errors in this printing of the proof. This makes what was a tall task virtually impossible.
So what did Godel do that was so interesting?
He proved that there were certain arithmetical statements about whole numbers that were not provable but true.Read more ›
Anyone who wants to trace this proof is free to do so. Though the formal logic can be formidable, and must be learned before tackling the proof, only the basic structure is necessary and it is not difficult to learn. It is also necessary to know a little about prime numbers, specifically that every composite number can be decomposed into some unique group of prime factors. Otherwise, all the technical aspects of the proof (barring the conclusion of theorem VI and a bit of the recursion) can be perfectly understood by someone outside of the world of formal mathmatics. The proof itself is meant for a professional mathmatician. If you are interested and willing this will not dissuade you. To say Godel was not a master of exposition is misleading for he is ,if nothing else, just that. I have heard working through the proof compared to a mystical experience and the proof itself to a symphony. It is truly beautiful to even the mere math enthusiast. Godel is not, however, a college professor and does not wish to explain what need not be explained. This will not be of much consolation when he prefaces a statement with, "of course," for the twentieth time and you have no idea what he is talking about. But if you are not afraid to go ahead when you have tried and failed to understand, and are not afraid to return when you have gained some small piece of the puzzle and try again, everything will come clear. This is the original. All the commentaries are great, and some are even helpful before you get to the conclusion, but they are not the proof and should not be taken as a substitute. They do not suffice the way a generic drug does. There is no way to understand the full scope of the proof if you are not willing to immerse yourself in it and the language it uses.Read more ›
This book is a translation of one of the most important papers in 20th-century mathematics. It's wonderful that Dover has published it at such a cheap price, so everyone interested in the incompleteness theorems can take a look at it. However, I should warn potential readers that it is _not_ the best introduction, for three reasons: (1) Goedel was not the world's greatest expositor. (2) We now have nearly 70 years worth of insight Goedel didn't have when writing this paper. (3) Goedel never intended the paper to be read by anyone but professional mathematicians. Non-mathematicians who really want to understand this material should also take a look at "Goedel's Proof" by Nagel and Newman (and perhaps Hofstadter's "Goedel, Escher, Bach: An Eternal Golden Braid" for cultural background). Mathematicians can find lots of more technical expositions. The original paper should not be the only source one tries to learn from, but I think it can be very valuable to take a look at it side-by-side with more modern treatments to get a feeling for how the ideas really arose. In principle one could learn everything straight from the source, but it just isn't the most efficient way. (I say this as a professional who has read the original paper and lots of other accounts of the proof, as well as written one of my own.) Net recommendation: this book is so cheap that one should buy it and a modern treatment.
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THE proof as Goedel wrote it (plus typos). I have seen modern proofs of this theorem which are much easier to follow (as an example, a Mir book on mathematical logic by a Russian mathematician whose name I cannot recall), but this one is the REAL thing. Modern proofs can be much clearer, but the original always has an added value. The writing style is not the best, but by reading this version you get a clearer idea of how Goedel came up with his theorem and the many difficulties he faced. Remember, by the time most of us read or heard about this for the first time, mathematical logic had advanced quite a few decades.