/lit/ - Literature

Kind of a shit post but I'll bite.

In his words, "GEB is a very personal attempt to say how it is that animate beings can come out of inanimate matter. What is a self, and how can a self come out of stuff that is as selfless as a stone or a puddle? What is an "I", and why are such things found (at least so far) only in association with, as poet Russell Edson once wonderfully phrased it, "teetering bulbs of dread and dream" - that is, only in association with certain kinds of gooey lumps encased in hard protective shells mounted atop mobile pedestals that roam the world on pairs of slightly fuzzy, jointed stilts?"

>>9707916
>Godel's incompleteness theorems are about self-reference
quick and dirty tip: if someone thinks this, you can tell that they only have a superficial "pop-STEM" understanding of the subject and don't know how the arithmetization of syntax works

>>9708229
it's like when biology teachers/profs talk about an adaptation's "purpose" in scare quotes -- the Godel sentence "talks about itself" in scare quotes
I think the practice of telling a false but digestible story as an introduction to a complicated topic is called "lying to children"

>>9708229
so what are godel's incompleteness theorems about, then?

I feel like I almost worked out what Godel was getting to, then I got too old. I tried though, when I still had enough neurons. And I got close enough to know that someone, sometime, will get there. There's something really big and really essential about it. It's part of The Thing that we haven't quite figured out yet.

Deriving philosophy from science.

>>9709542
>big and essential
>The Thing

Care to enlighten us about what you did find before you got too old?

>>9709556
Godel's self referentialism is somehow connected to consciousness. Godel numbering is definitely connected to the logic of how symbology relates to reality. How "words" connect to "things" in the real world. There are words that describe things and there are words that describe the words that describe things and there are ways to number those words as an abstraction. I can't work it out, but I honestly think someone will make the connection. Sorry if I am too obscure, but I can almost feel it.

>>9708328
The first incomplete theorem shows that a formal system into which arithmetic is adequately formalizable contain sentences that can neither be proved nor refuted. Since there are 'undecidable' sentences, the relevant systems are 'incomplete.'

The proof for the first theorem involves a sentence that many popularizers will say says "I am not true" or more accurately "I am not provable." But saying this is misleading, since the sentence in question is actually the numerical encoding that corresponds to the sentence "this sentence is not derivable" and not that sentence itself. As far as I can see, you lose any claims to self-reference once you do the encoding. And the proof proceeds on a merely syntactical basis (Godel even calls the initial encoding process "the arithmetization of syntax"), so I don't see why you'd be talking about reference at all unless as a didactic shorthand. At any rate, he shows that if the sentence is derivable, then the formalization of its derivability should be derivable. But given the nature of the sentence, you can show from its derivability the derivability of the formalization of its non-derivability. So you have a contradiction. You get a similar contradiction if the negation of the sentence is derivable (i.e., the sentence is refutable). So, assuming the system is consistent (i.e., for any given sentence not both it and its negation are derivable in the system), the sentence is neither provable nor refutable -- so, it's undecidable, and the system is incomplete.

The second incompleteness theorem shows a formalization of consistency of the system cannot be proven in the aforementioned formal system -- assuming that the system is in fact consistent. I won't go into many details here. Essentially, Godel comes up with a formalization of consistency. It turns out that if you can derive that formalization, you can then derive the Godel sentence used in the first theorem. But if you can derive that, as above, you get a contradiction. So, again assuming the system is consistent, the formalization of its own consistency is not derivable in it.

It's all just messing with syntax in formal systems and reference enters nowhere but post-hoc. It's scope seems pretty limited to mathematical logic.

>>9709644
No worries, thanks for sharing. I suck at math but definitely sounds intriguing, I can believe the obsession.

picked this up at the library the other day. It looked like a textbook inside so I put it right back down. Should I read it?

>>9709672
Thanks anon

>>9709542
Read this
https://arxiv.org/html/nlin/0004007

>>9707916
Of its consistent it's incomplete.
If it's complete it's inconsistent.
Move on.

>>9709833
I think so. It's one of my favorite books and it continues to award re-reading.

>>9710686
Sorry. Meant 'reward re-reading'.

All Mathematicians are Numerologists and troglodytes.