How Gödel's Proof Works (2020)

How Gödel's Proof Works (2020)(quantamagazine.org)

40 points by tzury 2 hours ago | 25 comments

pfdietz 14 minutes ago |

You can also obtain incompleteness from the unsolvability of the halting problem, by noting that if every statement in (say) Peano arithmetic were provable, one could solve the halting problem. Encode a halting execution of a TM as an integer using Gödel numbers and write a statement that the execution halts. Either that statement or its negation would be provable, so search for proofs for each at the same time.

matherial 1 hour ago |

> 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.

czgov 44 minutes ago | |

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.

jojomodding 7 minutes ago | | |

Yes it is true in the unique model of second order PA, aka the computable model, aka the "standard" model.

gavinsyancey 1 hour ago |

If you find this interesting, I highly recommend reading "Gödel, Escher, Bach: an Eternal Golden Braid"

andyjohnson0 1 hour ago | |

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.

gus_massa 24 minutes ago | | |

I really like GEB.

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.

bordo 1 hour ago | | |

I can second this as a wonderful introduction to the proofs. This is the book that got me into logic and formal methods.

undershirt 1 hour ago | | |

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.

trescenzi 1 hour ago | |

“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/

khazhoux 59 minutes ago | |

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.

riffraff 14 minutes ago | | |

I would say "never actually finished", but I think this is quite true.

quaverquaver 1 hour ago | |

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...

bobson381 58 minutes ago | | |

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

dsego 22 minutes ago |

I am wondering if I'm just not smart enough to understand, but I've managed to slog through GEB and in the end the proof seems contrived, it stands on self reference.

somethinsfishy 46 minutes ago |

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

3abiton 30 minutes ago | |

Even more videos to binge before the holidays end! Thanks for sharing!

manesioz 47 minutes ago |

Great breakdown: https://stopa.io/post/269

smfjaw 1 hour ago |

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

Paracompact 30 minutes ago |

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...

ChrisArchitect 58 minutes ago |

(2020)

Some previous discussions:

2023 https://news.ycombinator.com/item?id=38391787

2020 https://news.ycombinator.com/item?id=23832087

bananaflag 1 hour ago |

(2020)