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Comment by vbtemp | Hacker News Reader

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 few books I recommend:

"Incompleteness - The proof and paradox of Kurt Gödel" by Rebecca Goldstein

"Gödel's Proof" by Ernst Nagel (it's a tiny book, not too technical, but technical enough for anyone with a solid CS background to appreciate and understand)

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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.

Thanks! I believe I read a reference to this book once on HN, then could not find it. Unless there is another book that focuses on the misuses of the incompleteness theorem, this is the one I'm looking for.

Interesting. I'll look into that. THanks.

The Goldstein book is rubbish. Soloman Feferman destroys it in his LRB review.

http://www.lrb.co.uk/v28/n03/solomon-feferman/provenly-unpro...

http://math.stanford.edu/~feferman/papers/lrb.pdf (full text)

"Those who are fascinated by Gödel's theorems—and the general idea of limits to what we can know—may still hunger for a more universal view of their possible significance. But they should not be satisfied with Goldstein's 'vast and messy' goulash, hers is not a recipe for true understanding."

I read this stuff in "The Emperor's New Mind" by Roger Penrose. It not only talks about this, but introduces to Relativity, Quantum Mechanics, Phase Spaces. It also builds up all the mathematics required, so if you know what complex numbers are - that will be more than enough to follow and understand the concepts introduced well enough. Very highly recommended.

The second part somehow was not as interesting though.

"Emperor" is a problematic work at best. He's skyhooking.

I wouldn't say it's rubbish - but it's no great authority on communicating the proof itself, certainly.

It is a good biography though, covering the roots in the Vienna circle and the disagreements with Wittgenstein.

Edit: Thanks for posting that pdf though, I enjoyed reading it.

If it's a biography you want, you'd be better off with John Dawson's one (he is, of course, the author of the article linked at the root).

Another book that has several chapters related to GEB is David Deutsch's "The Beginning of Infinity." It's a very accessible read, and for me, eye-opening in more than one way.

+1, a really enjoyable but provocative read. With respect to Godel, Roger Penrose's "Shadows of the Mind" gives a pretty good insight into the implications of Godels theorems.

I have no background in CS or Math, but a lot of philosophy. In other words, I'm a highly interested layman. What's my best plan of action to understanding Godel's theory? Maybe the best approach would be an entry level book on CS?

Do you have a decent understanding of first order logic? If not, sort that out first. Peter Smith has a good list of books to choose from:

http://www.logicmatters.net/2012/05/teach-yourself-logic-1-f...

Then get Smith's book An Introduction to Gödel's Theorems. It explains Gödel's results in great detail and doesn't assume too much background knowledge. A certain amount of perseverance will, of course, be required…

There's some supplementary material on his website: http://www.logicmatters.net/igt/

At the very least a good course in discrete mathematics is a good start (it's also a good start for anything technical as well - one of the most valuable math classes anyone can ever take, as far as I'm concerned)

Following that, a good class in the theory of computing: understanding what exactly a generative grammar is, properties of classes of languages (e.g., understanding what "regular languages are closed under complimentation" means), pumping lemma, diagnalization proofs, halting problem. The incompleteness theorem is intimately tied to this. This is the "CS-route" to getting a good understanding in Incompleteness, I'm sure math or physics majors come to approach it in each their own way.

Being a little blunt, a background in philosophy (whether it's academic or not) without a solid discrete math background, doesn't help you out at all. This isn't philosophy, it's just a fact about properties of formal systems of sufficient complexity. If you're looking for philosophy you won't find anything too deep in the proof of Incompleteness. The philosophical implications are not clear.

However, I do recommend Rebecca Goldstein's book. It's not technical, and she's a Princeton philosopher who will indulge you with possible philosophical ramifications of the theorem (along with a good narrative). I also recommend her other books as well, especially her first novella "The Mind-Body Problem". From a philosophical perspective, the dispute between Goedel and Wittgenstein who never accepted the Incompleteness Theorem "whereof we cannot speak we must pass over in silence", which, ironically, speaks of something of which we cannot speak.

> At the very least a good course in discrete mathematics is a good start

I believe that the best starting point to get to incompleteness is formal logic. This is the basic set of concepts that lets us make terms, statements and finally proofs the subject of formal mathematical study, thus tying the loop (formally mathematically defined reasoning about formally mathematically defined reasoning :-) ) that leads to Goedels proof.

Discrete mathematics is helpful but it is rather low level, the core concepts in incompleteness come from formal logic.

I consider a solid discrete math curriculum to provide a reasonable background in first and second order logic

I would be surprised to find second order logic discussed in an introductory discrete math curriculum.

Knowing computer science stuff like regular languages is really not required to understand the incompleteness theorems; grasping first order logic, a little arithmetic and some elementary recursion theory is sufficient.

I repeat my objection to the Goldstein book raised elsewhere in this discussion.

I always liked "Godels Theorem Simplified". It doesn't rely on heavy technical prerequisites in mathematics or CS. It is pretty much as advertised, a simplification of Godel's original proof. Godel used a more complicated encoding scheme using prime numbers, which Gensler replaces with a simpler encoding scheme. He walks you through various less powerful formal systems, before you get to one complicated enough to have incompleteness issues. There is also discussion about the philosophical ramifications of Godel's theroems.

http://www.amazon.com/Godels-Theorem-Simplified-Harry-Gensle...

"Godel, Escher, Bach" is another interesting read, but that volume does have a lot of extraneous fluff.

extraneous fluff

Hey, now. Gödel, Escher, Bach has character and is IMO a very fun book. You might have to read it more then once, though.. it's self-referential and strange-loopy in that way.

It's an excellent book, but it's very broad, and requires a commitment in time.

If you want to understand just Godel's theorem, you probably want to focus on Logic more than Math or CS. Godel's proof involves qualifying over first-order logic statements and logical properties of numbers much more than it involves computation or numerics. The most nitty-gritty part is Godel encodings, but it's not that computationally intensive, and the details actually aren't that relevant.

If you want to understand the ramifications of Godel's theory, it impacts math more directly than CS. The most directly impacted branches of math are the more "fundamental" ones like Set Theory. Limitations of CS has a lot more to do with Turing's theorems than Godel's.

Actually Godel's proof is important to CS, it's an analogue of Turing's Proof concerning undecidable problems.

My understanding is that Godel's proof was the inspiration for much of Turing's work.

The unsolvability of the halting problem has a slightly weaker version of Goedel's first incompleteness theorem as a trivial corollary. You can use Turing machines to prove a stronger statement very easily as well. Therefore the CS perspective is very enlightening. See http://www.scottaaronson.com/blog/?p=710.

I would recommend coming to terms with "A Transition To Advanced Mathematics". Once you feel like you can plumb Cantor's Diagonalization without nausea :), then read on Godel. GEB is excellent but frothy ( not pointed and a slog ) but is worth it for other reasons.

I think that Godel was more important from a social perspective - there is a strong correlation between Postivism and much of the evils in the 20th Century.

I think it's best to go directly at the source. Wikipedia has a very understandable sketch of proof (which in itself is quite unconventional) that helps understand the motivation and the content of the theorem: http://en.wikipedia.org/wiki/Proof_sketch_for_G%C3%B6del%27s...

Whatever approach you take, be sure it includes Torkel Franzén's Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. Although it's pretty good, it may not be where you want to start. But there are a lot of bogus conclusions made by people with a superficial understanding of Gödel and that is the best thing I know pointing out the flaws.

Shameless self-promotion: I gave a talk outlining the proof. http://youtu.be/j0NcE3Tnklw

It was aimed at math students, but I don't think I assumed prior knowledge of anything esoteric. Parts of it might be hard to follow without math or CS, I'm not sure.

The statement "this fact is not provably true under the axioms of mathematics" is not provably true under the axioms of mathematics. Therefore, it is true, but we can never prove it.

I recommend approaching this via uncomputability, and the fact that for every program there is a proof and vice versa.

If you want to start off easy listen to this: http://www.radiolab.org/2011/oct/04/

My favorite book about incompleteness is Raymond Smullyan's Godel's Incompleteness Theorems.

I've always liked GW Flake's writing on the topic.

http://mitpress.mit.edu/books/FLAOH/cbnhtml/excerpts1.html

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