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Its always fascinating to read about Gödel. I have not read GEB, yet reading about his findings has really changed the way I think about things. Thanks for posting this article.
http://news.ycombinator.com/item?id=3991687
Mentioned in glowing terms here on HN many times:
http://www.hnsearch.com/search#request/all&q=logicomix
http://news.ycombinator.com/item?id=846451
http://news.ycombinator.com/item?id=870762
http://news.ycombinator.com/item?id=874471
http://news.ycombinator.com/item?id=3690254
It was my present for proofing an early draft of "Here's Looking at Euclid" / "Alex's Adventures in Numberland"
http://en.wikipedia.org/wiki/Oskar_Morgenstern
Morgenstern and Einstein were Gödel's closest friends, I've just now learned. It gives me goosebumps looking at that photo and imagining the three of them on that lawn.
Semi-related, here's an account of Gödel's "pent-up lecture" about the inconsistencies in the American constitution that he told to his citizenship examiner: http://morgenstern.jeffreykegler.com/
When I first became fascinated with incompleteness (following initial coursework in theory of computation), it kind of became my "religion" of sorts for a while. But as many mathematicians lament, the Incompleteness Theorem is one of the most popularly abused proofs of all time - used for non-experts to assert their own half-baked pseudo-philosophy (of course, the same goes for quantum mechanics as well). These are a…
When I first became fascinated with incompleteness (following initial coursework in theory of computation), it kind of became my "religion" of sorts for a while. But as many mathematicians lament, the Incompleteness Theorem is one of the most popularly abused proofs of all time - used for non-experts to assert their own half-baked pseudo-philosophy (of course, the same goes for quantum mechanics as well). These are a…
Nagel's book is a wonderful exposition (three-page-long footnotes aside). In addition to these two, I might recommend Torkel Franzén's book "Gödel's Theorem: An Incomplete Guide to Its Use and Abuse". Some of the content is fairly technical but not inaccessible by any means. If you're interested in the corner cases of how incompleteness theorems can be applied, it's a terrific resource.
What I get out of Goedel is this: There are some things that are true that cannot be proved.
This it more like it: For any consistent, finite axiomatized formal system that is sufficiently expressive (such as the Principia Mathematica), you can construct a sentence in the language of that formal system that asserts its own un-provability. Therefore, there does not exist a mechanistic method for enumerating over all true statements in the language of that formal system.
By stating "there are some true things that cannot be proved" goes too philosophy deep, and is outside of our pay-grade. Just consider: humans don't reason based on mechanistic principles - and there's no proof as to the expressability of natural language (though we can be sure it's aggravatingly inconsistent)
EDIT: I just want to say that in general, if someone does not really grasp the technical notion of a formal system, consistency, expressiveness, provability, soundness, or recursive enumeration, then it is basically impossible for them to appreciate the incompleteness theorems, and they are very likely to grossly misrepresent it.
What I get out of Goedel is this: There are some things that are true that cannot be proved.
GEB sits on my nightstand with too little time to be read. It might have to get bumped up the priority queue a bit.