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What Gödel Discovered (2020)

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Re: What Gödel Discovered (2020)

#11

Post by Nevermark »

dvt wrote: post #2

This blog post gets way too caught up in Gödel numbers, which are merely a technical detail (specifically how the encoding is done is irrelevant). A clever detail, but a detail nonetheless. Author gets lost in the sauce and kind of misses the forest for the trees. In class, we used Löb's Theorem[1] to prove Gödel, which is much more grokkable (and arguably even more clever). If you truly get Löb, it'll kind of blow y…

Other names for Gödel encoding: Digital. Binary. Zorros and Unos.

Today Gödel encoding is so pervasive, it’s easy to miss that everything is trivially Gödel encladed. Because like most everything invisible, it’s right in front of us.

We Gödel our memes and gift cards, and (pick your poison) pr0ns. Colors and AI’s, lax ASMR’s and our (sneaky don’t read me) terms of service. Even this very small humble .

Gödel isn’t eating the world. Gödel already pööped it.

Re: What Gödel Discovered (2020)

#12

Post by bazoom42 »

I’m confused by this jump to the natural world:

> could you encode in pure logic how a dog behaves

Assuming we knew enough about how a dog behaves (or less ambitiously, a more primitive organism) I would assume this could be described in a formal language. But why would Principia be needed for this? Math have been used to model natural phenomena a long time before Principia.

Re: What Gödel Discovered (2020)

#14

Post by adzm »

WhitneyLand wrote: post #13

Saying his name like “girdle”, is the closest English pronunciation I’ve seen. The actual German ö is hard for me to figure out without having a native speaker around to practice with.

I try to explain it as trying to shape your mouth as if you were saying e but say o instead.

Re: What Gödel Discovered (2020)

#15

Post by ergonaught »

bazoom42 wrote: post #12

I’m confused by this jump to the natural world: > could you encode in pure logic how a dog behaves Assuming we knew enough about how a dog behaves (or less ambitiously, a more primitive organism) I would assume this could be described in a formal language. But why would Principia be needed for this? Math have been used to model natural phenomena a long time before Principia.

> I would assume this could be described in a formal language

The assumption is the first problem, no? If the formal language is complete, it must be inconsistent. If it is consistent, it must be incomplete. If the language is incomplete or inconsistent, you may be unable to encode dog behavior in it.

Re: What Gödel Discovered (2020)

#16

Post by bazoom42 »

ergonaught wrote: post #15
bazoom42 wrote: post #12

I’m confused by this jump to the natural world: > could you encode in pure logic how a dog behaves Assuming we knew enough about how a dog behaves (or less ambitiously, a more primitive organism) I would assume this could be described in a formal language. But why would Principia be needed for this? Math have been used to model natural phenomena a long time before Principia.

> I would assume this could be described in a formal language The assumption is the first problem, no? If the formal language is complete, it must be inconsistent. If it is consistent, it must be incomplete. If the language is incomplete or inconsistent, you may be unable to encode dog behavior in it.

The article use “describing the behavior of a dog” as something people began to think was possible because of Principia. This is what I don’t get. Was this thought impossible before Principia? On what grounds?

What about describing an electrical circuit formally? Surely this was thought possible before Principia was published.

Re: What Gödel Discovered (2020)

#17

Post by roxolotl »

bazoom42 wrote: post #16
ergonaught wrote: post #15

Earlier quoted context omitted.

> I would assume this could be described in a formal language The assumption is the first problem, no? If the formal language is complete, it must be inconsistent. If it is consistent, it must be incomplete. If the language is incomplete or inconsistent, you may be unable to encode dog behavior in it.

The article use “describing the behavior of a dog” as something people began to think was possible because of Principia. This is what I don’t get. Was this thought impossible before Principia? On what grounds? What about describing an electrical circuit formally? Surely this was thought possible before Principia was published.

The assumption made by many in the early 20th century, spurred on by the recent successes of unification and formalization, was essentially that we could formally describe the entire universe. Godel’s proof shows that if you attempt to formally describe something there’s either an inconsistency or it’s incomplete. That doesn’t mean you cannot describe the behavior of a dog formally but it does mean the same formula which encodes the behavior will either be inconsistent or incomplete. It might only be inconsistent or incomplete when applied outside of defining the behavior of a dog though. That’s why the little preamble about unification exists in this post but it’s not very well tied into the rest of the post.

Re: What Gödel Discovered (2020)

#18

Post by qnleigh »

discarded1023 wrote: post #10
qnleigh wrote: post #6

Earlier quoted context omitted.

You might say "but wait, haven't we just proven that it's true? So isn't that also a contradiction?" This would be a disaster, because it would prove that the axioms of arithmetic are inconsistent! Now 1+1=3 for all we know. The catch is that when we proved that the sentence is not false, we used proof by contradiction, and for proof by contradiction to be a valid method of proof, we need to assume that the axioms we…

> We can't prove that the axioms of arithmetic are consistent [...] Sure we can! [1] ... but it requires (logically) stronger axioms. Assessing the relative strength of axioms along these (Gentzen's) lines goes by the name "ordinal analysis". It's not clear to me that stronger axioms are always less plausible than weaker ones (as axioms). An alternative is to abandon your insistence on consistency. Another thread poi…

> We can't prove that the axioms of arithmetic are consistent

using the axioms themselves. We can prove consistency using a stronger set of axioms, but those axioms have their own liar sentence, and so they can't prove their own consistency. And without knowing if the stronger set of axioms is consistent, we can't be sure that we have really proved the consistency of arithmetic.

Re: What Gödel Discovered (2020)

#19

Post by qnleigh »

eis wrote: post #9
qnleigh wrote: post #6

Earlier quoted context omitted.

You might say "but wait, haven't we just proven that it's true? So isn't that also a contradiction?" This would be a disaster, because it would prove that the axioms of arithmetic are inconsistent! Now 1+1=3 for all we know. The catch is that when we proved that the sentence is not false, we used proof by contradiction, and for proof by contradiction to be a valid method of proof, we need to assume that the axioms we…

> These theorems apply to any system of axioms that are rich enough to state the liar's paradox. Isn't that circular reasoning or tautological though? Rephrased: any system that can state something that these theorems apply to, can have the theorems applied to. I think the word "rich" is too inaccurate in this context. It is not clear why there can't be a more "rich" system which does not suffer from this issue and c…

Yeah rich is a vague word here. Really we're trying to say 'if a set of axioms can express provability, and can get a sentence to refer to itself, then it can state the liar sentence. And once it can state the liar sentence, then the rest of Gödel's argument follows.' There's no circularity.

Re: What Gödel Discovered (2020)

#20

Post by srcreigh »

WhitneyLand wrote: post #13

Saying his name like “girdle”, is the closest English pronunciation I’ve seen. The actual German ö is hard for me to figure out without having a native speaker around to practice with.

The “l” is just as hard to pronounce correctly. English has a very nasal “l” compared to german.

Try saying English “hell” but dragging out the “l” (“helllllllll”); very nasal. Compare audio here: https://en.wiktionary.org/wiki/hell#German

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