- Paperback: 160 pages
- Publisher: NYU Press; Revised edition (October 1, 2008)
- Language: English
- ISBN-10: 0814758371
- ISBN-13: 978-0814758373
- Product Dimensions: 5 x 0.4 x 8 inches
- Shipping Weight: 4.8 ounces (View shipping rates and policies)
- Average Customer Review: 97 customer reviews
- Amazon Best Sellers Rank: #114,825 in Books (See Top 100 in Books)
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Gödel's Proof Revised Edition
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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 --This text refers to the Hardcover edition.
"A little masterpiece of exegesis."-Nature
"An excellent nontechnical account of the substance of Gödel's celebrated paper."-American Mathematical Society
Top customer reviews
Godel numbering is a way to map all the expressions generated by the successive application of axioms back onto numbers, which are themselves instantiated as a "model" of the axioms. The hard part of it is to do this by avoiding the "circular hell". Russell in Principia Mathematica tried hard to avoid the kind of paradoxes like "Set of all elements which do not belong to the set". Godel's proof tries hard to avoid more complicated paradoxes like this :
Let p = "Is a sum of two primes" be a property some numbers might possess. This property can be stated precisely using axioms, and symbols can be mapped to numbers. ( for e.g open a text file, write down the statement and look at its ASCII representation ). The let n(p) be the number corresponding to p. If n(p) satisfies p, then we say n(p) is Richardian, else not. Being Richardian itself is a meta-mathematical property r = "A number which satisfies the property described by its reverse ASCII representation". Note that it is a proper statement represented by the symbols that make up your axioms. Now, you ask if n(r) is Richardian, and the usual problem emerges : n(r) is Richardian iff it is not Richardian. This apparent conundrum, as the authors say, is a hoax. We wanted to represent arithmetical statements as numbers, but switched over to representing meta-mathematical statements as numbers. Godel's proof avoid cheating like this by carefully mirroring all meta-mathematical statements within the arithmetic, and not just conflating the two. Four parts to it.
1. Construct a meta-mathemtical formula G that represents "The formula G is not demonstratable". ( Like Richardian )
2. G is demonstrable if and only if ~G is demonstrable ( Like Richardian)
3. Though G is not demonstrable, G is true in the sense that it asserts a certain arithmetical property which can be exactly defined. ( Unlike Richardian ).
4. Finally, Godel showed that the meta-mathematical statement "if `Arithmetic is consistent' then G follows" is demonstrable. Then he showed that "Arithmetic is consistent" is not demonstrable.
It took me a while to pour over the details, back and forth between pages. I'm still not at the level where I can explain the proof to anyone clearly, but I intend to get there eventually. Iterating is the key.
When I first came across Godel's theorem, I was horrified, dismayed, disillusioned and above all confounded - how can successively applying axioms over and over not fill up the space of all theorems? Now, I'm slowly recuperating. One non-mathematical, intuitive, consoling thought that keeps popping into my mind is : If the axioms to describe arithmetic ( or something of a higher, but finite complexity ) were consistent and complete, then why those axioms? Who ordained them? Why not something else? If it turned out that way, then the question of which is more fundamental : physics or logic would be resolved. I would be shocked if it were possible to decouple the two and rank them - one as more fundamental than the other. I'm very slowly beginning to understand why Godel's discovery was a shock to me.
You see, I'm good at rolling with it while I'm working away, but deep down, I don't believe in Mathematical platonism, or logicism, or formalism or any philosophical ideal that tries to universally quantify.
Kindle Edition - I would advise against the Kindle edition as reading anything with a decent bit of math content isn't a very linear process. Turning pages, referring to footnotes and figures isn't easy on the Kindle.
You should note that the Kindle Edition had some confusing formatting. For example, here is a screenshot of a page ([...]) where there is a severe lack of formatting for exponents. Should at least read "2^8 x 3^(11^2) x 5^2 x 7^(11^2) x 11^9 x 13^3 x 17^(11^2)", so that it is at least marginally easier to discern where the exponents are. Yes, you ought to be able to figure it out with a moment's thought if you're reading the text that accompanies it, but I thought it was a little disappointing and that it warranted some mention in a review. This has made me cautious of buying any further math or math-related books for my kindle.