Can someone explain Godel's Incompleteness Theorem is supposed to work? It says that soemwhere out there there's a number that we can't prove exists? but how do we know that it's really out there if we can't prove it?

In essence, given a set of axioms there will always exist some questions that can't be proven. However you don't know what those questions are.

that which is consistent, is incomplete
that which is complete, is inconsistent

When you ask to understand Gödel and struggle with it, please remember that you're into a complex part of the higher mathematics.
It's no shame to struggle with understand it. In contrasts, it' good that you choice this.

In my opinion, in order to understand Gödels proof, you need two points:
1.) Understanding Gödelisierung, that means that you translate logical (and extralogical) symbols like "->","^" and so on into numbers.
On this trick, you can translate entire proofes and theorems into a sequence of numbers.
2.) Cantorization.
You make use some kind of Cantor's diagonalization on this number codes.
Like Cantor, who proves that there are no bijuncitve relation between the set of natural numbers and real numbers.

I hope it helps.

P.S.: OP is a fag.

>>16692644
Thats just some kind of rephrasing in informal language that helps OP nothing.
>>16692652
Kek.

>>16692642 (OP)
it says our retarded mathematical systems are inconsistent.

>>16692652
if its complete but inconsistent then its not complete.

you just have 2 shit tier mathematical systems that you can barely use

>>16692642 (OP)
What's your math background? I always like breaking the theorem down into parts. Godels is about mathematical models, so let's apply it to ZFC set theory. As a warmup, let's consider a model with only a few axioms, say, empty set and power set. What sets can you prove exist? What sets can't you prove exist?

>>16693185
>it says our retarded mathematical systems are inconsistent.
are there not retarded systems that are consistent?

>>16694475
yes writing down every tally one by one, adding a zero doesn't even help

>>16692642 (OP)
Prease. Accept the mrystery.

>>16692642 (OP)
It's just a rehash of the liar paradox "this statement is false" in dumb math syntax

>>16692642 (OP)
>It says that soemwhere out there there's a number that we can't prove exists? but how do we know that it's really out there if we can't prove it?
In many cases, we have the formulas, just not the memory space to execute them take tree(3) factorial for instance.

>>16694909
No, it doesn't say "this is false", it says "this is incomplete".

>>16695027
no, it says "this is unprovable"

>>16695047
No, it says it can prove that not all values can be represented with decimal numbers or any other number system because "they are incomplete".

>>16692642 (OP)
One way of “understanding” the theorem and some of its proofs is:
>A Turing machines cannot solve the Halting Problem.
>There is a nice way of encoding Turing machines into [math]\mathbb N[/math]
>The Halting problem becomes a yes-or-no question about [math]\mathbb N[/math]
>Therefore a Turing machine cannot always answer these questions
The basic statement about complete and consistent axiom systems falls out of this. The only problem is the proof is longer when you expand all the steps formally, but conceptually I think it’s useful

>>16692642 (OP)
https://www.youtube.com/watch?v=mhIkyqLDl9M

Here, let me help you understand.

>>16694909
No, it is saying that not all statements can be solved with simple true or false conventions (or any other formal system of solution), so it would be like saying "these statements are neither true nor false".

>>16692642 (OP)
For every number you provide, I can give you +1 for a new number. There are natural numbers so unimaginably large we cannot even write them down. G(64,64) is grahams number. It's clearly in N, but absent any ability to even write it down, clearly we cannot formally prove it in N using the axioms of arithmetic.

>>16695027
>>16695814
"This sentence is false" is the textbook example of a statement that is clearly neither true nor false, and is a wonderful example of the incompleteness theorem . Stop being a sperg.

>>16696011
It is an example of something that can't be determined to be true or false, but incompleteness is not a rehash of that self-referential paradox, they are two different things, you are the tard conflating unrelated things.

>>16696021
A formal system is complete if for every statement of the language of the system, either the statement or its negation can be derived (i.e., proved) in the system.

The liars paradox is a very trivial counterexample and is this due facto in indicator of an incomplete formal system.

>>16692642 (OP)
Gödel demonstrated a method for converting sentences of predicate logic to numbers, and (more importantly) for converting the inference rules of predicate logic to functions. He then used this to construct a paradoxical statement: "there exists no proof of this statement" (similar to the Epimenides paradox, which is often paraphrased as "this sentence is false").

If the statement is false (i.e. there does exist a proof of it), then predicate logic is unsound: you have a statement which can be proven yet is false. If the statement is true, then predicate logic is incomplete: you have a true statement for which no proof exists. The reasoning can be generalised: any formal system capable of expressing its own metatheory must be incomplete. IOW, mathematics will never be "solved"; there will always be unsolved problems.

Essentially, paradox (or self-contradiction) undermines notions of absolutes. E.g. "can God create a stone which is too heavy for him to lift?".

>>16694909
It's not surprising that you can say self-contradictory things.

What is surprising is that if you have an axiomatic system that's complex enough for *arithmetic*, then you can twist it around to make statements about itself and you can produce things like the liar's paradox in it. AND that has a further implication that there are true statements about it that are unprovable within it that aren't paradoxes. Shocking, actually, if your were ever overly impressed with axiomatic methods in math.

>>16696099
>twist it around
You mean mapping the system of arithmetic into some different domain, where a whole different set of rules apply.. and then concluding the original system is in some way defective? This sounds like a horrible abuse of logic. But at least it's kosher in metalogic or whatever domain of reasoning this horse shit exists in.

>>16696116
A system that's sufficient to prove things about arithmetic is not exactly the same thing as "the system of arithmetic". And it's not "defective", it's just not capable of proving every valid statement within it - that's what "incompleteness" means.

>>16696164
Weird way with agree with me but okay. Dressing my post up in fancier language while saying you disagree is awk

>>16696006
Ok how about one hundred?

>>16696041
but godel's thing is that it is true but can't be proven in the system, are you saying that the liars paradox is true but can't be proven?

>>16696248
A common misunderstanding is to interpret Gödel’s first theorem as showing that there are truths that cannot be proved. This is, however, incorrect, for the incompleteness theorem does not deal with provability in any absolute sense, but only concerns derivability in some particular formal system or another. For any statement A unprovable in a particular formal system F, there are, trivially, other formal systems in which A is provable (take A as an axiom).

>>16696251
It doesn't seem like you understood the question, do you have a different formal system that can mathematically show that the liar's paradox is true, but can't be proven like godel did with his theorems or not?

>>16696064
>can God create a stone which is too heavy for him to lift?
This one is simple to solve, though.
God could just turn everything into the rock so that lift loses meaning because there is no outside of rock.

>>16696811
For any statement A unprovable in a particular formal system F, there are, trivially, other formal systems in which A is provable (take A as an axiom).

It should be called Godel's confusion because he wants "truth" and "proof" and "formula" to mean different things at the same time. He wants to use them in their proper meta-mathematical way (the same way all branches of math use them) and also cause confusion by redefining those terms according to mathematical definitions. This is just like taking the fictional world of a novel as if it were real or allowing a C compiler to use a reserved keyword as an identifier.
See p. 50 of the C99 standard, section 6.4.1 Keywords:
https://www.dii.uchile.cl/~daespino/files/Iso_C_1999_definition.pdf
The situation has never been cleared up. Books such as
Kunen - Set Theory: An Introduction to Independence Proofs
Enderton - A Mathematical Introduction to Logic
Jech - Set Theory: The Third Millennium Edition
Hodges - A Shorter Model Theory
all fall short of a hygienic approach that avoids reusing terms reserved for a meta-mathematical purpose for mathematical definitions. They all descend to vague, imprecise, confusing statements and hand waving. It's a disgrace.

>>16696988
In model theory, there's another error which is shocking in its dishonesty: claiming (without explanation) that an axiom schema can be replaced with an infinite collection of axioms. This is particularly nuts because meta-mathematics works by rewriting strings of mathematical symbols and mathematics works by applying inference rules to axioms. It is like saying that the text you read that is real and in front of your eyes is the same thing as the imagined fantasy world of sets and collections.
Oddly enough, this error echoes Plato when he asserts the real existence of Forms beyond the physical, material world of perception and sense.

>>16696993
we get it, you are an intuitionist

>>16696988
Im not trusting any C standard hosted on dot cl

>>16697038
I'm not an intuitionist.
>>16697079
See section 2.1 Variable Names
https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libros/%5BKernighan-Ritchie%5DThe_C_Programming_Language.pdf

>>16696860
You still don't seem to understand the question.
Which formal system proves the liar's paradox, A, if the true/false system which I assume you are referring to as F can't?

>>16697364
NTA but just consider a formal system which includes the liar paradox statement as an axiom.

>>16697382
Yes, that is the axiom in question that anon was suppose to be formalizing, now at least you maybe understand the question, so now you can go ahead and provide an actual answer instead of restating the question.

>>16697388
I just answered your question. You have some formal system in which you state the liar paradox statement and add an axiom to that formal system which states that the liar paradox statement is true.

>>16697390
>I just answered your question.
No, you restated my question.
>You have some formal system in which you state the liar paradox statement
Yes, the true/false system.
> add an axiom to that formal system which states that the liar paradox statement is true.
The liar's paradox statement says that it is false, though, so your axiom still contradicts it, you have not solved anything, you have just contradicted yourself again.

>>16697391
>The liar's paradox statement says that it is false, though, so your axiom still contradicts it
You are making a mistake here which is not distinguishing between syntax and semantics. It doesn't matter what the liar paradox statement "says" because that's semantics or in other words, it depends on your model for your axiom system. The new axiom only contradicts the liar paradox statement if you consider a model where the liar paradox statement is false, but you're not under any obligation to consider such a model in the first place.

>>16697395
>if you consider a model where the liar paradox statement is false,
The only written axiom of the liar's paradox is that it is false, so there is no other model to consider unless you can provide these other magical axioms that somehow make it not self-contradictory.

>you're not under any obligation to consider such a model in the first place.
It is literally the one and only obligation you have because the starting axiom is that it is false.

>>16697397
>The only written axiom of the liar's paradox is that it is false
No, in the original axiomatic system, the liar paradox statement is just a statement. There's no axiom where it is proclaimed to be true or false. You have to assume that the liar paradox statement is unprovable in this original axiom system. Now if you have a statement which is unprovable in some axiom system, then there is a model for the axioms where it is true and also a model where it is false. This is a godel completeness theorem. So if we assume that in our initial axiom system, it is unprovable, we get the existence of models where it takes any truth value you want. So you can add either of the axioms "the liar paradox statement is true" or "the liar paradox statement is false" and still get consistent systems.

>>16697401
>the liar paradox statement is just a statement.
The statement is the axiom, axioms are statements.
>There's no axiom where it is proclaimed to be true or false.
The only axiom of the system is that it is false, retard.
>Now if you have a statement which is unprovable
Its not unprovable, it self-contradicting.

>there is a model for the axioms where it is true and also a model where it is false
You keep saying this, but you clearly can't actually provide it and are too retarded to understand that what you are providing doesn't resolve any contradiction.

>"the liar paradox statement is true" or "the liar paradox statement is false" and still get consistent systems.
No, it is not consistent to say it is true and it is false because true/false are opposites, you have provide no such logic where they can be used otherwise.

>>16697403
> it self-contradicting.
You are failing to making the syntax/semantics distinction again
>it is not consistent to say it is true and it is false because true/false are opposites
It is consistent because we are talking about different models. Think of a multiverse. In one universe, you can be a virgin and in another universe you can be a non-virgin.
>but you clearly can't actually provide it
It's proved by the completeness theorem (although that might require assuming the axiom of choice iirc)

>>16697406
>It is consistent because we are talking about different models.
No, you have not redefined true and false in a way that they are not opposites, so your model still is self-contradictory because you are using contradictory logics to be equal.

> Think of a multiverse.
An axiomatic system is not a multiverse of completing axiomatic systems, it is a single logical system of determining truths and untruths called false.

>It's proved by the completeness theorem
No, according to incompleteness a completeness theorem would necessarily be completely inconsistent, so that system would be unable to determine truths vs untruths.

>>16697416
>completing
*competing

>>16697106
dot mx? really?

>>16697602
racists go to /b/

>>16692642 (OP)
>It says that soemwhere out there there's a number that we can't prove exists? but how do we know that it's really out there if we can't prove it?
You have hit with the common misconception pseuds have when trying to bring the theorem into discussion.

But correcting you first, you're obfuscating the theorem by saying there are numbers that we can't prove that exists.

The actual pseud claim is that there are true statements in any axiomatic system that can't be proven true within the system. That's a poorly formulated statement that as is, isn't true.

The actual result of the theorem is that there are statements that can't be proven true or false, that can be formed with the language used in the axiomatic system. It doesn't mean that there are things that are true and cannot be proven, it quite literally means that you can't prove it true or false, and that isn't a hole, it means that there are statements that no matter how you look at it it's not gonna be true in general unless you add an axiom that says it is false or it is true.

So your intuition that "how do we know that it's really out there if we can't prove it?" agrees with what I think too, the vague statement faggots make when bringing up this theorem doesn't really follow from this theorem, as from outside the axiom system, the statement can be made true or false.

One such case predicted by this theorem, is the choice axiom (seen as a statement, though it has "axiom" in its name), it's known that it cannot be proven true or false under a certain axiom system. Mind you that that wasn't proved with Godel's incompleteness theorem. Since Godel's theorem only tells you that any axiom system has those statements, it doesn't tell you how they are found.

Godel or whoever else might have originally formulated it with or changed it to be "numbers" instead of statements from formal logic, but that IMO obfuscates the real meaning.

>>16697779
No, the numeric encoding is actually necessary since it's what allows the axiomatic system to address the universe of possible statements (each encodable with a number).

>>16692642 (OP)
What you got to understand is how to fold math in on itself. I prefer a left-handed undertuck, myself but that's only because I use the Abelian school.

>>16697779
>Since Godel's theorem only tells you that any axiom system has those statements, it doesn't tell you how they are found.
Non-constructivist math was a MISTAKE

>>16697779
godel's thing ain't about independence, nigger

>>16698451
>independence
Ctr+F: None in my post.
Kill yourself schizo nigger.