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On Formally Undecidable Propositions of Principia Mathematica and Related Systems [Paperback]

Kurt Gödel
5.0 out of 5 stars  See all reviews (8 customer reviews)
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Book Description

April 1 1992 0486669807 978-0486669809 New edition

In 1931, a young Austrian mathematician published an epoch-making paper containing one of the most revolutionary ideas in logic since Aristotle. Kurt Giidel maintained, and offered detailed proof, that in any arithmetic system, even in elementary parts of arithmetic, there are propositions which cannot be proved or disproved within the system. It is thus uncertain that the basic axioms of arithmetic will not give rise to contradictions. The repercussions of this discovery are still being felt and debated in 20th-century mathematics.
The present volume reprints the first English translation of Giidel's far-reaching work. Not only does it make the argument more intelligible, but the introduction contributed by Professor R. B. Braithwaite (Cambridge University}, an excellent work of scholarship in its own right, illuminates it by paraphrasing the major part of the argument.
This Dover edition thus makes widely available a superb edition of a classic work of original thought, one that will be of profound interest to mathematicians, logicians and anyone interested in the history of attempts to establish axioms that would provide a rigorous basis for all mathematics. Translated by B. Meltzer, University of Edinburgh. Preface. Introduction by R. B. Braithwaite.

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On Formally Undecidable Propositions of Principia Mathematica and Related Systems + Godel's Proof + The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine
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3 of 3 people found the following review helpful
5.0 out of 5 stars A profound paper, but difficult to read Sept. 26 2000
By A Customer
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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3 of 3 people found the following review helpful
5.0 out of 5 stars From the horse's mouth, 'le text' Aug. 2 2003
Speaking not as a math specialist but one disposed to read a number of the popular explications of Godel's famous proof I can say that it was Godel's original text that did it for me. The reason is that it is the proof and not a lot of verbiage about the proof. And it is short and sweet. One problem is that the more common Turing Machine approach is actually 'easier', where Godel's approach is that of recursive functions which are more obscure, or at least less often discussed. If you can sort of glare at the recursive function issue and proceed with the basics of the proof it will stand out suddenly better than many of the popularizations. At least give it a try.
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2 of 2 people found the following review helpful
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.
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1 of 1 people found the following review helpful
5.0 out of 5 stars Read the masters! Oct. 31 2001
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.
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