gödel's incompleteness theorem

[Anonymous No.15918736]

I don't get it

[Anonymous No.15918875]

Sedevantist wolf

[Anonymous No.15919895]

what book is this

[Anonymous No.15920123]

>>15918736
>I don't get it
Because the proof is incomplete

[Anonymous No.15920448]

>>15920123
and left up to the reader as an exercise

[Anonymous No.15920696]

>>15918736
Don't waste time with that crap, it uses a diagonal argument therefore is nothing but useless nonsense.

[Anonymous No.15920698]

it looks like shit, OP

[Anonymous No.15920715]

>>15918736
It's just a fancy halting problem

[Anonymous No.15920732]

>>15920696
Bertrand is that you? You got a better definition for the cardinality of infinite sets? You can't escape self reference by pretending it doesn't exist when the whole goal of formalized systems is for them to prove themselves consistent.

[Anonymous No.15920809]

>>15918736
If you want to grok it, read Gödel Escher Bach. If you just a breakdown of the logic of the proof, read this:https://plato.stanford.edu/entries/goedel-incompleteness/
Basically, any formal system of logic robust enough to prove itself consistent is robust enough to express the statement 'This statement has no proof'. If you can prove it, the system is inconsistent. If you can prove you can't, the statement is true, and then there are true statements in the system that can't be proved, making it incomplete. All of our understanding of math and logic is based on expressions in these systems. It might be that the only unprovable true statements are piddly selfreferential technicalities like that one... but we haven't found any way to prove that. There might be more. The Riemann hypothesis might be true and the proof for it simply does not exist. If you're invested in proving it though you can't dwell on that possibility. Camus says “There is only one really serious philosophical problem, and that is suicide. Deciding whether or not life is worth living is to answer the fundamental question in philosophy. All other questions follow from that.” Gödel's incompleteness theorem could be said to pose a similarly fundamental question of mathematics, whether it is worth trying to prove anything as true. If you say yes, carry on with the faith your proof is out there. If not, forget about mathematical truth and just try to make the most of your finite uncertain existence. Or kill yourself and answer both questions in one go

[Anonymous No.15921031]

>Peano arithmetic is suppose to specify the natural numbers
>There is a statement phi which PA can never prove neither disprove
>Either phi or not phi is true when interpreted with regards to the natural numbers (trivial)
>There are true statements about the natural numbers which cannot be proved in PA
>Because PA is first order, and first order logic is complete, it means there are deranged models for PA restricting what PA can prove
>Using a different system besides PA doesn't fix the issue

t. mathematician

[Anonymous No.15921730]

>>15920809
>formal logic systems are self-referential
Tarski's truth meme. His idea was you can apply this to math and define a function "eval" that evaluates the truth of a sentence. Now, the only problem is that Tarski never defined truth in the first place. It's like a communist / totalitarian meme or joke, truth is whatever the authority says truth is, haha, and it can change arbitrarily after the fact because it was never defined!
This is how commies think of math. They think it's just another toilet for them to shit in.

[Anonymous No.15921765]

https://archive.org/details/TheOriginsOfTotalitarianism/page/n369/mode/2up
pic related
Posting in a /lit/ thread that got lost, went through a bout of amnesia, convinced itself it was a /sci/ thread, got taken in by an occult math symbol cult somewhere in LA (and, let's face it, LSD is probably going to be involved at some point) then got covered in tattoos and wound up here
all because Tarski is the greatest memer of all time
>LSD and communism is groovy, right?
right. the nice men in white jackets will help you. remember to take your meds, Anon

[Anonymous No.15921959]

>>15920696
>it uses a diagonal argument therefore is nothing but useless nonsense.
you know the proof for the halting problem uses a diagonal argument, right?, are you declaring that halting is decidable?

[Anonymous No.15921965]

>>15921730
>truth is whatever the authority says truth is
nah, people like you make me glad that paulo freire did what he did

[Anonymous No.15921988]

>>15921965
Oh no, you don't understand at all; Tarski and Godel are treating "truth" as a free form creative writing contest. It's all about style and tradition. Godel/Tarski are math fascists. They're letting some fundamentally toxic material in that is going to screw things up later. Godel and Tarski are hacking math, and they're arrogating totalitarian privileges such as ruling by decree. It's The Wizard of Oz, Math Club style.

[Anonymous No.15922012]

>>15921988
The error that makes this grafting of poetry and philosophy onto mathematical logic possible is the introduction of a "truth evaluation function" that has the property of giving the truth value of sentence, i.e. it just magically "knows" the laws of manipulation logical formulas even though we don't have to specify what those are...
if you smell snake oil, it's because you're standing in a puddle of it
Godel / Tarski / Enderton have cheated you out of a manifest of the rules of logic.

[Anonymous No.15922015]

>>15922012
what's so screwed up is that intuitively you KNOW that you haven't been provided with the inference rules for classical logic, yet the author carries on as if such a list of rules had already been provided

[Anonymous No.15922018]

>>15922015
A summary of the inference rules for classical logic start on p. 573
link: https://homepages.uc.edu/~martinj/Symbolic_Logic/341%20Syllabus,%20Textbook,%20Handouts,%20Notes/LPL%20textbook.pdf
Nothing like this appears in Enderton, yet Enderton carries on as if these rules had been provided. This is all because Tarski is playing word games with math terms and nobody is catching him, they're just falling into the same degenerate behavior.

[Anonymous No.15922020]

>>15921730
Let's hear your nonaxiomatic definition of truth then. You can't just say that which is provable in first order Zermelo-Fraenkel set theory. That's doing the same thing

[Anonymous No.15922021]

>>15921730
>Now, the only problem is that Tarski never defined truth in the first place
the model theoretic notion of first-order truth is literally named after him

[Anonymous No.15922022]

>>15922018
Tarski's word game is so slick you can't believe it isn't math. However, it's pure wordplay and trickery. You're getting caught in a trap, you aren't actually studying a mathematical object.
Beautiful, but deadly.
https://en.wikipedia.org/wiki/Truth#Tarski's_semantics
>'P' is true if and only if P
where 'P' refers to the sentence (the sentence's name), and P is just the sentence itself.

[Anonymous No.15922028]

>>15922020
we don't define truth because sets aren't formulas
in reality, we don't even the language of proof to math students or graders, they just intuitively apply what amounts to a proof checker for information Fitch derivations with nearly zero awareness of the historical process that produced this standard practice
TL;DR we don't define truth, we pass it along by convention, and you just have to pick up, say, the scoping rules for guard variables even though nobody has ever heard of that or knows what you mean in the context of informal math proofs graded by homework graders

[Anonymous No.15922030]

>>15922021
see
>>15922022
Tarski simply ignores the possibility that the rules of logic haven't been stated yet
this is what you get with Bolsheviks
they're retards

[Anonymous No.15922062]

>>15921959
Are you declaring that infinite is finite?

[Anonymous No.15922067]

>>15922030
not talking about that definition of truth, talking about this one
https://en.wikipedia.org/wiki/T-schema#The_inductive_definition

[Anonymous No.15922086]

>>15922067
https://www.youtube.com/watch?v=qN5zw04WxCc
The Who - My Definition

[Anonymous No.15922093]

>>15918736
its just complicated cope

[Anonymous No.15922097]

>>15921959
halting is decidable

[Anonymous No.15922104]

>>15922097
how?

[Anonymous No.15922119]

>>15922097
if the halting is decidable, wouldn’t that mean the collatz conjecture is true?

[Anonymous No.15922120]

>>15922104
Exclude programs that include the halts function

[Anonymous No.15922233]

>>15922120
then you can't decide all halting problems

[Anonymous No.15922234]

>>15922028
Pass it along from where? What are the conventions? If you were to formalize them, would they start to look like... axioms? If you can't formalize them and they're more like porn ie. 'you know it when you see it', how is that different from 'truth is whatever the authority says truth is'?