/lit/ - Literature

>>15445530

Yes, it is not intuitive nor make any sense, that is why it was ground breaking at the time, but things are the way they are.

There are some easy examples, just google it. I'm almost sleeping now, got back to see if I find some thread about Ouroboros and Socrates but it was archived already.

I need a paper that popularizes it the best

>>15445530
As far as I understand it (which I don't), Gödel created a way to map certain numbers with statements such as "x is red" or "all triangles have three sides." From this, he tried to create a system whereby axioms could be represented by such numbers and statements that logically follow from certain axioms would be their product. He then proposed the statement "this statement cannot be proven true by the provided axioms," which, if assumed to be false, means that the statement can be proven and is thus true, but cannot be proven.
I've also heard that the conclusion of the incompleteness theorem can be summarized by saying that every mathematical system is either inconsistent/contradictory or incomplete (meaning that it does not account for all true statements).
Keep in mind that I am not knowledgeable about this and everything I've said is probably unsubstantial compared to Gödel's actual methodology.

>>15445667
that's pretty much the gist of it, or what I at least got from it.
that's what I was saying about 'true statements', how can they really be true in a sense if they are not forward-deduced

There are two incompleteness theorems, and the second follows from the first.
The first says that if a set of axioms is self-consistent (i.e. does not lead to contradictory proofs), there are necessarily statements within that system (specifically about the natural numbers) that can be constructed using those axioms that CANNOT be proven, regardless of whether they are actually true or not. The second theorem says that the self-consistency of the axiomatic system in question cannot be proven with its own axioms. There is a really good video I found on it months ago that I will post when I find.
t. graduate student in mathematics, for whatever that's worth
>>15445667
yes

>>15445530
>>15445696
https://youtu.be/V49i_LM8B0E?t=4960

For every axiomatic system that is consistent, there are necessarily some statements which cannot be proven (incomplete).
For every axiomatic system that is complete, there are necessarily inconsistencies.
Hence,
> (1) This statement cannot be proven
If (1) is false, then it can be proven. However the proposition states that it cannot be proven. This is a contradiction.
If (1) is true, then it cannot be proven, thus the system is incomplete.
This is the quickest rundown.

>>15445686
I guess that because when we take the statement "this statement cannot be proven by the axioms" to be false, it turns out to be true but not provable, there may exist some class of statements which are likewise true but unprovable. The only problem is that we could never prove that they are both true and unprovable besides said statement. In fact, this specific statement could be the only example of something being both true and unprovable or it could be a look into a massive set of other statements. Some people have hypothesized that things like the Riemann hypothesis could be one of these statements, but it is impossible to prove that it is both true and unprovable, the best we could do is one or the other.
What's interesting to think about is that if it is shown to be true that the Riemann hypothesis cannot be proven by the peano axioms, that may give more credence to the claim that it is true, as we have yet to find an exception to the hypothesis.

>>15445696
>necessarily statements within that system
that helps, now I only need to realize how these statement are even possible to construct.
since the coding numerizes axioms, first thing I thought was statements like this:
+%5&*=+
but doubt it would simple as that.

>>15445530
It means that arithmetic doesn't describe arithmetic. You need another system to describe why arithmetic works. You can add, multiply, divide the cardinal numbers together in a predictable way - it works, but to describe why that it works requires another system, and that system itself needs another system. Hence the internal incompleteness of axiomatic systems, and the incompleteness is that there's always extra rules you need to explain the rules.

Big Dick Schuller explains it at 1:26:50

https://www.youtube.com/watch?v=V49i_LM8B0E

>>15445767
>>15445709
>>15445696
Great job anon(s)

>>15445530
I genuinely cannot understand how this cannot be understood

>>15445530
It's like saying you can't prove math within math you'd need something prior to it that is in an asymmetric relationship with it, like logic, to prove the axioms of math

>>15445763
comprehensible. so there is inference from unprovable true statements to the ability to derive description (I wont try to describe in other words)

>>15445761
Watch the youtube video I linked, it takes about 10 minutes for him to get there, but he describes how this can be done using meta-mathematical statements

>lecture
yep, I know it was the same uncountable set trick once again.
so the best way to put it: show that this set of axioms consistent with the this set of axioms.

>>15445667
>Gödel created a way to map certain numbers with statements such as "x is red" or "all triangles have three sides."
i think this should be
Gödel created a way to map certain statements such as "x is red" or "all triangles have three sides" with numbers.

>>15445696 yeah?

>>15445696
reminder that 'true'' in logic has nothing to do with truth and true in casual language

>>15446764
t. moron

>>15446787
t. undergrad

>>15446764
qualify your assertion

Can we talk about this subject more? This implies our physical universe is incomplete. Knowledge is then also incomplete. It is not therefore not possible to know everything because there is no everything to know.

>>15446977
>This implies our physical universe is incomplete.
no it doesnt

>Knowledge is then also incomplete
duh

>It is not therefore not possible to know everything
also duh

>>15447005
>no it doesnt
Huh? Why do you say so? If our physical universe was complete then wouldn't knowledge also be complete?

>>15445667
>summarized by saying that every mathematical system is either inconsistent/contradictory or incomplete (meaning that it does not account for all true statements).
Sufficiently complex systems, but yes. It's a formalised "the answers you seek only pose more questions".

Read Godel, Escher, Bach by Hofstadter if you're not a brainlet

>>15447031
first you have to explain what you mean by physical universe and what it means to be complete.
for example, if i hold an apple in my hand the godel theorem doesn't imply that apple is incomplete. godel is describing the incompleteness of the reference system we use to communicate 'completeness', ie we know the apple is complete because it is an apple. and we know it's an apple because we know it's an apple.
now turn 'apple' into 'x': we know x is x because x is x. he's pointing out the fundamental self-referenciality of mathematics using mathematics to describe mathematics.
does that make sense?

>>15447031
Completeness is a property of formal systems.

>>15447057
Stop pretending you have any idea what you're talking about.

>>15447063
that's exactly what he's talking about. he's pointing out you have to use fundamental mathematics to justify why mathematics 'works'. he's pointing out the axiomatics system's inherit use of self-reference.

if you disagree, state why and qualify your answer, otherwise you're just saying "nuh uhhh"

>>15446977
Bertrand Russell & co. btfo.

>>15447051
>not
How did that get in there? Whoops!

>>15447075
There are complete systems capable of self-reference.

>>15447092
You're talking about simple systems, right?

>>15447092
i didn't say anything to indicate otherwise.

>>15447092
>he's pointing out the axiomatics system's inherit use of self-reference.

>There are complete systems capable of self-reference.

nice reading comprehension

>>15447095
>>15447103
>>15447106
Reread the thread. You're not making any sense.

>>15445530
He's stating that you cannot have completeness on a formal system capable of expressing basic arithmetic.
Aside from btfo some retarded theorists of the early 20th century (like Bertrand Reddit) the brainlet rundown is that mechanical applications of terms shuffling in propositions will never ever make mathematics.
It also entails a refutation of any form of 'finitism' in representing formal systems (let alone thought in general). Godel considered it a refutation of materialism for instance.

>>15447125
i'm not making any sense to you because you don't understand what godel was talking about. either offer your interpretation and we can discuss it or keep looking like a retard.

>>15447095 isn't me

>>15447138
You have no background in logic or mathematics, and are talking out your ass.

>>15447133
You're an idiot.

>>15447142
cool, so you're just not going to offer a counter argument.

>>15447145
Another brilliant post.

>>15447151
hey, i'm the dude >>15447142 is replying to. can you give me the source where godel considered it a refutation of materialism, i'd be interested to read that

>>15447133
wtf but wildberger told me sideways 8 was a spook

>>15447057
Sure it makes sense. I mean by completeness theorem is that it's like a ruler.

If we use an incomplete system that doesn't know "everything" then the things that we measure are also "incomplete". I am only entertaining this idea because I think it would be an interesting topic.

>>15447138
I think the first anon was a troll
>>15447063
Either way you correctly assume that he isn't me.


I'm going to put up a name so people don't impersonate me. These are my posts:
>>15446977
>>15447031

>>15447280
>> If we use an incomplete system that doesn't know "everything" then the things that we measure are also "incomplete".
It does not follow at all. Formal systems are a specific type of structures that in no way exhaust knowledge.
The conclusion is correct by stretching 'completeness' outside its definition for formal systems though and Godel would have approved, empirical objects are never given adequately.

>>15445530
I highly recommend Godel's Proof by Ernest Nagel to understand better. It explains the meta mathematics really well.

bump

>>15445686
>>15445530

hmmm true but not provable... true but not provable...

oh I know! It's an axiom!

The incompleteness theorem means there are infinitely many axioms and the best analogy I've heard has to do with primes.

Think of each prime number as an axiom of the set of all natural numbers, that is to say, all natural numbers are some linear combinations of primes, just like all provable statements are the consequence of axioms. The point is you can never have the set of all axioms of provable statements, because there are infinitely many, just as there any infinitely many primes.

>>15445530
Suprisingly good takes itt >>15445667, >>15445696, >>15445709.

To add more specifics and dispel common confusions :

1. All theorems have a range of application. The incompleteness theorem, only applies to first-order logic system that contain Peano arithmetic (usual arithmetic). A smaller system of arithmetic (like Presburger's arithmetic, which is Peano arithmetic without multiplication) is not concerned by it, nor is a completely different system (like the theory of totally ordered complete fields with characteristic 0). Both systems are complete and the former can even prove its own consistency (not sure about the latter). There are other examples.

2. Therefore the incompleteness theorem doesn't apply to all of logic, all of mathematics much less so all of knowledge (mathematicians make suprisingly little use of logic btw). It's really a specific (if important) statement about formal logic.

3. A proposition that cannot be proven in a system might nonetheless be provable in another system. For instance Peano arithmetic cannot prove its own consistency, but a larger system that includes Peano arithmetic might prove the consistency of Peano. However that larger system would have its own unprovable propositions. The consistency of Peano arithmetic was indeed proved by Gentzen, who relied on a transfinite induction that isn't allowed in Peano arithmetic.

A few asides that might be interesting :

4. Gödel also proved a completeness theorem: any classical system in first-order logic is coherent (it cannot prove a proposition and its negation at the same time) if and only if it is consistent (it has a model, ie a mathematical structure that embodies its true propositions). So for a first order system, internal consistency by demonstration = external consistency with a mathematical structure. This is why consistent is often used interchangeably with coherent in those discussions. Note that the completeness theorem doesn't contradict the incompleteness theorem.

5. There's a similar but somewhat wider theorem called Tarski's undefinability theorem. If you want to talk about the incompleteness of human knowledge it's a better starting point that Gödel's, but that would still be stretching it a bit.

6. Gödel proves his theorem by constructing an unprovable proposition. It is rather formal and contrived (essentially a formal version of "this proposition cannot be proved"). There are less artificial arithmetically unprovable proposition that were discovered later, see Goodstein's theorem. They can still be proven using non-arithmetic means.

6. Apart from changing the system of axioms, one can weasel a way out of Gödel's limitation by allowing different rules of demonstration. Gödel's work mostly applies to classical logic, but there is intuitionist logic, minimal logic, modal logic, etc. All those make sense in their own realms, but you have to give up some things and accept others.

>>15447133
Worth noting two things on top of that:

1. Hilbert and co focused on elementary (finitist) propositions, and those representations alone can't represent arithmetic. However ci-elementary propositions can. The issue is that the kind of proposition with think of as "nice", "simple", "fondamental" and "uncontroversially true" produce issues, while it is entirely possible to derive all of arithmetic using less natural-sounding sets of propositions. Essentially there is a discrepancy between our intuition of what should be logically sufficient and what is mechanically sufficient in formal logic.

2. Most of arithmetic can actually be proved by elementary means (see Gentzen's Cut elimination theorem). However there's no clear bound on the size of demonstrations. So for instance, there is a demonstration of Fermat's last theorem that only relies on algebraic manipulation of integers, but that demonstration might require more symbol than there is atoms in the universe. This, again, shows that our intuition that "elementary" methods are better is fundamentally misguided.

>>15449105
*co-elementary propositions (ie the set of propositions that aren't elementary)

>>15449054
based post

>>15449105
are you a finitist ?
do you believe in induction?

>>15449164
I make no ontological commitments when it comes to mathematics. Why should I get riled up that someone uses big number? Induction is a fine tool for a mathematician, only über-autist get caught up on whether the product of that induction "actually exists".

>>15448727
Nope, that ain't it chief. Even with infinitely many axioms, no formal system can prove all the truths of elementary arithmetic.

>>15449244
Glad to hear maths are a meaningless game of symbols.

>>15445667
>>15445696
>>15445709
>>15449054
Based thread. Math + lit = CHAD.

>>15449290
Math is a language. What you use for is another story that can't be told in a mathematical language.

this thread at least proves bell curve to be complete

>>15449281
If a statement is true but unprovable, then is it not by definition an axiom?

>>15449988
The problem is that such a set of axioms would not be recursively enumerable.

>A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set.

>This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic and Zermelo–Fraenkel set theory (ZFC).

>The theory known as True Arithmetic consists of all true statements about the standard integers in the language of Peano arithmetic. This theory is consistent, and complete, and contains a sufficient amount of arithmetic. However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.

https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Effective_axiomatization