I am wondering if I'm just not smart enough to understand, but I've managed to slog through GEB and in the end it the proof seems contrived, it stands on self reference.
> However, although G is undecidable, it’s clearly true.
That's... not really true; it's surprising to see it in Quanta, of all places.
Godel's (separate) completeness theorem says that in first-order logic, anything that's semantically true in all possible scenarios can be syntactically proved. So, if G is "clearly true", that ought to make it provable.
The theorems don't contradict each other because in FOL, G is not guaranteed to be true. Its truth is independent of the machinery Godel put in place.
It's not something you really need to get into an introductory text, but it actually makes the whole outcome easier to grasp, and leads to many more counterintuitive results, such as Skolem's paradox.
It’s clearly true in The Natural Numbers. It’s not provable because in some model it’s false. Being clearly true in one model does not make it provable.
GEB is a great book, and I've probably read it at least 3.33333333... times over the years. As a late teen it blew my mind. But I'm not sure I'd recommend it as a route into Gödel's proofs [1]. The book covers a lot of other ground too, and is notoriously digressive and quirky (looking at you, dialogues).
Instead I'd recommend Gödel's Proof by Nagel and Newman for a conceptual intro.
[1] I'm not a mathematician, so my understanding is necessarily informal.
Most proof of the Gödel theorem use the primes encoding that is makes all the operations very unintuitive. But GEB uses just ascii and a lot of the side task get obvious. (It uses base 20 instead of 256, but it's the same idea.)
> is¨notoriously digressive and quirky
It is super mega ultra notoriously digressive and quirky.
I’ll never understand how GEB was using math, art, and music to explain consciousness (and Hofstadter himself still thinks no one understood it), but Nagel and Newman did a great job explaining why logic as a mechanical thing has only a tenuous relationship to concepts we understand, and that helped me crack at least a little bit of the mystery I was after when giving up on GEB.
I have not read GEB but I thought his second book, I am a Strange Loop, did a pretty good job of connecting the idea of self referential loops (like in godels proof) to consciousness and art and such.
It’s actually an interesting fact that every person who was programming in the 80s owns a copy of GEB, which they put on the bookshelf and never actually read.
while it is a groovy into to recursion and other cool ideas, GEB annoys me in that I feel like the three figures in the title are ill matched. Godel proves a super important result in math, sure... Escher was a skilled draughtsman who had a feel for tesselation. An OK artist IMO but no special insights. Bach on the other hand was an expressive genius who in the volume, power and beauty of his productions just seemed to drop out the sky like a meteor. Escher does not belong in the same breath frankly. if Bach made a crab canon or did other marginally math-y things that is just not the point - the work lives or dies in entirely different terms...
also,
"Gödel's incompleteness theorems: The proof that broke mathematics" | Joel David Hamkins
https://www.youtube.com/watch?v=Sza69An_H8o
spam-bait title but excellent mid-level talk.
I think Godel's theorem is the single most important result in mathematics. At the same time, when the subject comes up, I like to link people to this essay to dispel a lot of the woo surrounding it regarding human exceptionalism, religion, etc:
This is my favourite proof in all of maths (that I've been exposed to). Truly unreal feeling proving a statement is unprovable using godel numbering in an exam
That's... not really true; it's surprising to see it in Quanta, of all places.
Godel's (separate) completeness theorem says that in first-order logic, anything that's semantically true in all possible scenarios can be syntactically proved. So, if G is "clearly true", that ought to make it provable.
The theorems don't contradict each other because in FOL, G is not guaranteed to be true. Its truth is independent of the machinery Godel put in place.
It's not something you really need to get into an introductory text, but it actually makes the whole outcome easier to grasp, and leads to many more counterintuitive results, such as Skolem's paradox.
Instead I'd recommend Gödel's Proof by Nagel and Newman for a conceptual intro.
[1] I'm not a mathematician, so my understanding is necessarily informal.
Most proof of the Gödel theorem use the primes encoding that is makes all the operations very unintuitive. But GEB uses just ascii and a lot of the side task get obvious. (It uses base 20 instead of 256, but it's the same idea.)
> is¨notoriously digressive and quirky
It is super mega ultra notoriously digressive and quirky.
https://nyupress.org/9780814758014/godels-proof/
(It's the title of his follow up work after GEB.)
Joel David Hamkins - Oxford lectures on the philosophy of mathematics "The Gödel incompleteness phenomenon" https://www.youtube.com/watch?v=Y5trjR5aw0k
also, "Gödel's incompleteness theorems: The proof that broke mathematics" | Joel David Hamkins https://www.youtube.com/watch?v=Sza69An_H8o spam-bait title but excellent mid-level talk.
edit: speling
https://shs.cairn.info/revue-internationale-de-philosophie-2...
Some previous discussions:
2023 https://news.ycombinator.com/item?id=38391787
2020 https://news.ycombinator.com/item?id=23832087