How Gödel's Proof Works (2020)
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How Gödel's Proof Works (2020)
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Re: How Gödel's Proof Works (2020)
#2(2020)
Re: How Gödel's Proof Works (2020)
#3If you find this interesting, I highly recommend reading "Gödel, Escher, Bach: an Eternal Golden Braid"
Re: How Gödel's Proof Works (2020)
#4If 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.
Re: How Gödel's Proof Works (2020)
#5If you find this interesting, I highly recommend reading "Gödel, Escher, Bach: an Eternal Golden Braid"
“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).
Re: How Gödel's Proof Works (2020)
#6If 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 m…
I can second this as a wonderful introduction to the proofs. This is the book that got me into logic and formal methods.
Re: How Gödel's Proof Works (2020)
#7If you find this interesting, I highly recommend reading "Gödel, Escher, Bach: an Eternal Golden Braid"
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...
Re: How Gödel's Proof Works (2020)
#8This 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
Re: How Gödel's Proof Works (2020)
#9> 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.
Re: How Gödel's Proof Works (2020)
#10If 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 m…
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.