/sci/ - Science & Math

>>4321326
No, nobody here as read the entire incompl

if you just want to understand it, the result follows fairly trivially from the halting problem. But no, I haven't read it. I don't even speak German.

>>4321344

any read it at all?

i began reading it but i didn't complete it

>>4321326
Yes. Any linear system cannot be completely proven, only circular systems of logic can prove themselves and therefore be completely proven.

i've read the paper up to the proof of incompleteness. I haven't read the part on consistency. it is surprisingly simple considering that it is such an important result.

>>4321356
That's not what it says at all.

>>4321326
If we have a formal system, one that can at least express the notion of arithmetic, then there are true statements that cannot be proven within the formal system, UNLESS the formal system is not sound.
The statement "this formal system is sound", will be in that class of statements, hence, if it's true, it cannot be proven.

>>4321370
>That's not what it says at all.
>proceeds to say exactly what I said, but showing less understanding and more wikipedia

Yeah, cool story, bro.

>>4321372
It has nothing to do with circularity, and the notion of linear system doesn't even exist.
(Unless you mean a linear equation system, which is completely unrelated.)

>>4321372
I haven't even looked at wikipedia, but if you mean that they use the same keywords, then that's not a coincidence at all. Statements similar to mine are simply the most concise way of putting it. One needs the notions of soundness; and the fact that it applies only to sufficiently powerful systems is also relevant.

>>4321377
>>4321381
Y'know, I don't even care.

If you guys want to believe these two fags be my guest. Be advised, they will spew large amounts of bullshit.

>>4321386
>I don't even care because I am a faggot
Who knew??

The actual paper (or at least the English translation) is an absolute nightmare. Just get a basic undergraduate logic text (Leary, Enderton, Mendelson, etc) if you want to understand it. It's pretty fun stuff when it's broken down for simpletons like myself.

>>4321408
I've studied the simplified proofs in mendelson, smullyan and other books. i might make videos for youtube that go through the original paper line by line.

i ignored the dude who said it is a statement "about linear systems." whats a linear system? I've studied this for awhile and i have not come across that phrase.

The proof seems entirely precise and not "a nightmare" at all. There are mistakes in most english translations so im referring back to the german. (for example there should be a negation sign over xB(17genr) in equation 15

>>4321481
Ah, you've actually had a serious attempt at comprehensing the whole thing. I gotta be honest with you, I haven't actually read the original. I have had a course where the incompleteness theorem was one of the central pillars, so I have read modern proofs of the incompleteness theorem.
Honestly, I think you should just focus on proofs in textbooks, it contains the same information, but then formatted for readability. The first paper on a subject is always famous for it's concepts, not it's readability.

>>4321511

the odd thing is i don't find the original paper all that difficult. i actually find it much simpler. it is much more to the point. I think that most people get stuck on Godel's system P and questions about why it contains Principia and why he uses the notation he does. But I'm simply taking P for what it is; a series of symbols that generate new sequences of symbols. The key part of the text is the proof that the system can represent recursive functions. Once you admit that, then the result is obvious because xBy is recursive. So, the proof is easy if you think of it as a trivial application of the theory of recursions

>>4321529
I agree with that, but I find it difficult to interpret the grand idea with his notation. It works better if I can keep track of the semantics, while he's doing his symbolic manipulations.

bamp

Sometimes I come here and wonder what the fuck ANY of you are talking about. Then I get impressed and feel inferior. Then I go back to /ic/ where I fucking belong and were people can admire what I do instead.

Like seriously what the fuck is going on.

>>4321809
Science. Science is what the fuck is going on.

what's all this philosophy doing on /sci/?

>>4321809

can't speak for the other threads, but this one is about Godel's proof that in any mathematical system, strong enough to include arithmetic, there will be unprovable propositions. The proof is no longer considered advanced, but in 1930 some great mathematicians really struggled with it.

>>4321827
science = good philosophy

>>4321820

It's not philosophy, it's math.

bamped