> 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.
Right, but the whole point is that the axioms don't solely describe natural numbers. Godel's incompleteness is not a theorem about natural numbers, it's a theorem about their approximation in first-order logic.
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.
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
> 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.
Right, but the whole point is that the axioms don't solely describe natural numbers. Godel's incompleteness is not a theorem about natural numbers, it's a theorem about their approximation in first-order logic.
If you find this interesting, I highly recommend reading "Gödel, Escher, Bach: an Eternal Golden Braid"
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.
I can second this as a wonderful introduction to the proofs. This is the book that got me into logic and formal methods.
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.
“Godel’s Proof”[1] is also a great and shorter read if the scale of GEB is intimidating(I know it was for me at first).
https://nyupress.org/9780814758014/godels-proof/
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...
the linking thread for all three is self-reference, either in the form of a fugue or in a painting showing its own creation. Doug is a loop guy
A Strange Loop guy, to be precise.
(It's the title of his follow up work after GEB.)
If you like video, supplement your reading with
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
Even more videos to binge before the holidays end! Thanks for sharing!
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:
https://shs.cairn.info/revue-internationale-de-philosophie-2...
Great breakdown: https://stopa.io/post/269
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
(2020)
Some previous discussions:
2023 https://news.ycombinator.com/item?id=38391787
2020 https://news.ycombinator.com/item?id=23832087
(2020)