/sci/ - Science & Math

>>11624275
The implications are that there are some mathematical statements that are true but can not be proved

>>11624293
Dumb and wrong.
>>11624275
One implication is that math is not merely a mechanical game of formula-shuffling. Hilbert was wrong.

>>11624275
>tfw you will never have a theory of everything

>What I call the theological worldview is the idea that the world and everything in it has meaning and reason, and in particular a good and indubitable meaning. It follows immediately that our worldly existence, since it has in itself at most a very dubious meaning, can only be means to the end of another existence. The idea that everything in the world has a meaning [reason] is an exact analogue of the principle that everything has a cause, on which rests all of science. [217]
Pop sci atheist cult BTFO

>>11624308
>Dumb and wrong.
Undecidable statements exist, wtf are you talking about

>>11624356
Sure, CH is an undecidable statement. This is irrelevant.

>>11624308
Would this imply that at least Mathematical Platonism is true of reality?

>>11624371
the CH is true and unprovable. How is this irrelevant

>>11624381
It implies nothing of the sort, no.

>>11624384
>the CH is true
Wrong.

>>11624387
Not wrong, the Continuum hypothesis is true and unprovable. This is reality.

>>11624275
mathematicians don't study truth; OP is bogus

>>11624389
>CH is true
How do you know that?

>>11624275
>ontological
>>>/x/

>>11624389
>>11624384
Retard or trolling.

>>11624385
Was not Gödel a Platonist himself??? Sorry I am just trying to understand what would follow from this, I understand the theorems but what would the theorems imply about all of maths?
>>11624400
>durrr im a closed minded retard

>>11624389
Simply wrong, CH has been shown to be independent of the rest of mathematics. You may aswell take its falsehood as an axiom.

>>11624469
One of the theorems implies that we never know whether our current axiomatic system is consistent. We can try to show it using more powerful systems but then we wouldn't know whether that system is consistent so maybe it's not and then our current one isn't either. It's a bit of a gamble but we're pretty certain they're consistent, at least there's no merit from pretending they're not.

The other theorem has way more important implications, since it shows that no finite axiomatic system is equivalent to true algebra. Therefore there may be some hypothesis like the unsolved goldbach conjecture for which no proof is devisible in our current axiomatic system.

>>11624275
>using "3" as existential quantifier
if I didn't know godel already I would have no idea what this is saying

>>11624275
>What are the ontological implications of these theorems?
There are none. If any, there are epistemological implications.

>>11624487
>It's a bit of a gamble but we're pretty certain they're consistent, at least there's no merit from pretending they're not.
So, then there are no implications and this is a massive waste of time to even consider.

>"This statement is false!"

>>11624275
>What are the ontological implications of these theorems?
You are probably gay, but there will never be a formal proof demonstrating that you are gay

>>11624335
Based Godel, how will Reddit ever recover?

>>11626000
You're thinking of Tarski's theorem there, not Gödel's. Nice trips though.

>>11624469
i viewed it as potentially platonic too, "statement X has a value even if we are unaware of it"

Notes on this questionable thread:

CH is a bad example to make the point, since that is merely independent of (some popular) set theory. There's more hands-on examples, e.g. about naturals.
Btw. I think Gödel thought (as a Platonist) c=|N_2|.

"True but unprovable" is a shitty formulation of Gödel as well.
What we can say is that assuming all the metalogical presumptions to do Peano arithmetic (or other systems with particular conditions on their axiomatic formulation), there's statement about naturals that neither have a finite proof, nor do their negation.

There's nice models that validate both statements (but of course an example is not a proof of the general case)

Okay now if you really want, you could adopt LEM in the meta-logic to make the case that one of the two undecidable statements is "true" but what does that mean or do?

As far as consistency is concerned, Gödels arithmetizised consistency statement is not the only way of framing Hilbert's quest and there's model logic proof as well as completely different proofs using ordinals, that respectively both say that Peano arithmetic is consistent. I.e. it would be a dare to be against the consistency.

>>11628133
*modal logic proofs of consistency (not model, a typo)

can someone expplain me the proof? i just dont get it

>>11628157
Encode logical language in arithmetic.
Rough idea:
>OP=5, is=7, faggot=11.
>Exponentiation=infix sentence structure
>7^(5*11)=7*7*7*7*7*7*7*7*...*7*7 (that's some number with 45 digit)
Each sentence (like "OP is a faggot) gets a unique Gödel numnber this way due to arithmetical properties (that large as number can be factored into sevevens and the amount (55) can be factored into 5 and 11, i.e. you can connect language (logic language in Gödels proof, about arithemtic) with arithmetical expressions themselves.
He encodes proofs of arithemtic as arithemtical formulas.
Then he finds a statement that claims one thing in the encoded fashion and it's negation in what's encoded. A variant of "the 'statement' is unprovable" where "statement" represents the whole four word sentence in encoded fashion.

It follows if you could prove the statement that encodes, then what you proven was that you can't, an inconsistency.
So, if arithemtic is consistent, that whole statement can't have a proof.

>>11628157 #
He shows how to encode logical language in arithmetic.
Rough idea:
>OP=5, is=7, faggot=11.
>Exponentiation=infix sentence structure
>7^(5*11)=7*7*7*7*7*7*7*7*...*7*7 (that's some number with 45 digit)
Each sentence (like "OP is faggot") gets a unique Gödel numnber this way due to arithmetical properties (that large as number can be factored into sevevens and the amount (55) can be factored into 5 and 11, i.e. you can connect language (logic language in Gödels proof, about arithemtic) with arithmetical expressions themselves.

He also encodes proofs of arithemtic as arithemtical formulas/equalities.

Then he finds a statement that claims one thing in the encoded fashion and it's negation in what's encoded.
A variant of "the 'statement' is unprovable" where "statement" represents the whole four word sentence in encoded fashion.

It follows if you could prove the statement that encodes, then what you proven was that you can't, an inconsistency.
So, if arithemtic is consistent, that whole statement can't have a proof.

>>11628216
hey man thanks for this, i really appreciated it.

>>11624731
You misunderstand what he wrote. Godels theorem implies that which he wrote, that any axiomatic system we use is a gamble.

>>11624397
He's an infinitard true believer.
Infinityfags literally think like this.

>>11628296
CH has nothing to do with the axiom of infinity.

>>11626592
OP is a faggot, we take this as axiomatic

>>11628300
>CH doesn't involve infinity
You're a fucking retard.

>>11628300
That's a stretch, but I think nobody spoke about that axiom in particular here anyway

>>11628311
>>11628317
Still, believing in CH has nothing to do with whether or not one believes in infinity.

>>11624275
If someone ever asks for proof that all mathematicians are either schizos or autistic, point them to Gödel as an example of both.

>>11628329
I'm open for discussion.
For starters, I don't think nor "believes in infinity" nor the opposite is a precondition to do formal logic or math.
I suppose people who think CH ist Capital T true will also believe in infinity, whatever that really means.
What's your guys discussion even about?
Sounds like not a fruitful discussion.
Go out or do math, it's nice day

>>11628216
all of it made sense until
>prove the statement that encodes, then what you proven was that you can't, an inconsistency.
>So, if arithemtic is consistent, that whole statement can't have a proof.
what does "prove the statement that encodes" mean? a proof about the encoding algorithm? about the encoded statement? What about it?

and so what if the whole self referential statement cant have a proof? why is it a big deal? (what are we trying to prove about it anyway?)
sorry if its dumb im genuinely confused not trolling

since we're on this thread i feel like i have a related problem to godels problem, its bugged me for 2 years now, can anyone help?
>the object with no traits
the issue i have with it is referring to it at all gives it the trait of being referred to, making it exist, but how it can it be referred to if it doesnt exist (has no traits)?

>>11628396
to elaborate i feel like this is worse than godels problem or undecidability problems because this is to me what appears to be a genuine performance of a contradiction, not merely noting a contradiction that cant occur but actually witnessing reality (in thought) contradict itself

furthermore, discussion of this problem actually turns out to be either fruitless to an outsiders perspective or incoherent/impossible to an insiders perspective, because discussion and meta discussion of it contradict the objects nonexistence (--what i just did was meta discussion)

>>11628329
>Still, believing in their pot of gold has nothing to do with whether or not one believes in leprechauns.
You have no idea what you're talking about.

>>11628396
wait a second, i realize my error
>the thing that isnt
>the a that != a
okay, so thats flopped from the outset

how about nothingness though? it obeys similar issues - not a thing yet noting it makes it one,yet it cant be. mere contemplation of nothingness is a contradictory, yet this occurs on a daily basis as nothing is a word in the english language. what should one make of this?

>>11628396
what do you call an object?
what do you call a trait?

>>11628396
That sounds a lot like Berry paradox
https://en.m.wikipedia.org/wiki/Berry_paradox
Maybe you find hints in the discussion section.
>>11628396
It's moreover a matter of discussion to what extent you want to grant descriptions pinning down entities in the first place, see e.g.
https://en.m.wikipedia.org/wiki/Theory_of_descriptions
But that's philosophy proper, not math.

>>11628396
That sounds a lot like Berry paradox
https://en.m.wikipedia.org/wiki/Berry_paradox
Maybe you find hints in the discussion section.
>>11628396
It's moreover a matter of discussion to what extent you want to grant descriptions pinning down entities in the first place, see e.g.
https://en.m.wikipedia.org/wiki/Theory_of_descriptions
But that's philosophy proper, not math.

>>11628396
That sounds remeniscent of Berry paradox
https://en.m.wikipedia.org/wiki/Berry_paradox
Maybe you find hints in the discussion section there.
It's also up to debate to what extent we want descriptions even let narrow down entities in the first place, as in
https://en.m.wikipedia.org/wiki/Theory_of_descriptions
But in any case, that's philosophy proper, not math.

>>11628386
>and so what if the whole self referential statement cant have a proof? why is it a big deal?
It translates to a statement in arithemtic that has no proof (and neither it's negation).
Yes I suppose it's less surprising / a big deal today where we know universal computation is possible (and ten year olds code up while loops) than it was upon it's discovery in 1936. Note that Turings "invention" of the principles of a computer in the 30's was a side effect on continuing Gödels Work in the 30's, that was a possible answer to Hilbert's question around 1890.

cont

>11628396 #
That sounds remeniscent of Berry paradox
https://en.m.wikipedia.org/wiki/Berry_paradox
Maybe you find hints in the discussion section there.
It's also up to debate to what extent we want descriptions even let narrow down entities in the first place, as in
https://en.m.wikipedia.org/wiki/Theory_of_descriptions
But in any case, that's philosophy proper, not math.

>>11628386 #
>and so what if the whole self referential statement cant have a proof? why is it a big deal?
It translates to a statement in arithemtic that has no proof (and neither it's negation).
Yes I suppose it's less surprising / a big deal today where we know universal computation is possible (and ten year olds code up while loops) than it was upon it's discovery in 1936. Note that Turings "invention" of the principles of a computer in the 30's was a side effect on continuing Gödels Work in the 30's, that was a possible answer to Hilbert's question around 1890.


The encoded and encoding statements here are all themselves in correspondence to statements in the proof theory of arithmetic. The "the equation a=b has a proof" here becomes a statement of the form c=b itself.
It's a hard proof and if you want technicalities, you might just have to get into the technicalities.
E.g. one ingredient he cooked up to get at it is
https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_%CE%B2_function

For those interested, there's a book by, I think, Smith on the subject.

>>11628438
i think the defining of object itself can pose issues. i think this question hinges around existence vs nonexistence and how to formalize it, because at the end of the day existence and nonexistence are labels. one could hypothetically make other "instances" ex-instance, nonex-instance, 3-instance, twel-instance, etc. so thats not relevant

if we say
>object is a member of set with trait existence (E)
then we're doing something like assuming presence of members, and then how do we even begin to discuss members?
>item "members" is in set with E
does that work? i dont see an issue

as far as trait,
>traits are in the set of "attached to objects"
>traits are objects because above is in E
>objects are not traits but every object has a trait, they occur 1 to 1, to a non platonist theyre indistinguishable

so my prior thing seemed to be erroneous. I was generating a contradiction and getting worked up
>the trait haver with no traits
is what i said

but how about this?
>nonexistence
i just referenced nothing

so nothing automatically blanks your mind unless you assume the contradiction of nothing existing
^^^^^^
the above statement can be read two ways

>there are no things which cause mind blanking, mind does not blank, mind remains unblank
>some "thing" called "no-thing" causes blanking
but the second is an erroneous implicit assumption

>>11628406
What you said is completely unrelated to Gödels theorems. It's an amusing thing to think about because it shows how self-referential language gives rise to problems. It's essentially the same thing as "this sentence is a lie".
>>11628409
Read >>11628349
You can believe in CH and not believe in infinity. Actually, if you don't believe in infinity then you MUST believe in CH. The fact that someone believes in CH does not tell you anything about whether or not they believe in infinity.
You're a retard.

>>11628475
>You can believe in CH and not believe in infinity. Actually, if you don't believe in infinity then you MUST believe in CH. The fact that someone believes in CH does not tell you anything about whether or not they believe in infinity.
You're a retard.
I don't follow what you mean by believing in CH here.

I suppose what we can all agree on is that one way of speaking about about CH is as a string in the logic of set which, and one that in the comkon proof-theoretical frameworks has been found to be underivable from "good" axioms, where "good" is social notion that roughly comes down to "consicely expressible and interpretable in the language" (e.g. the axiom of pairing is a nice concise axiom with a clear interpretation, while L=V in terms of \in is actually unweildly since it makes use of ordinals that are not primitive as seen from the other common axioms).
From a pure formalist perspective that I've sketched so far, CH is not only a 140 year old question that a few decades ago has been found to be unprovable from the standard axioms, but even since has there not been "good" axioms other than the claim itself that would decide it one way or another. Afaik.

So that out of the way, what's a notion to "believe" in CH that can be held without also believing in the constituents of it's formulation (e.g. at least aleph null, i.e. an infinity)?

>>11628475
>You can believe in CH and not believe in infinity. Actually, if you don't believe in infinity then you MUST believe in CH. The fact that someone believes in CH does not tell you anything about whether or not they believe in infinity.

I don't follow what you mean by believing in CH here.

I suppose what we can all agree on is that one way of speaking about about CH is as a string in the logic of set which, and one that in the comkon proof-theoretical frameworks has been found to be underivable from "good" axioms, where "good" is social notion that roughly comes down to "consicely expressible and interpretable in the language" (e.g. the axiom of pairing is a nice concise axiom with a clear interpretation, while L=V in terms of \in is actually unweildly since it makes use of ordinals that are not primitive as seen from the other common axioms).
From a pure formalist perspective that I've sketched so far, CH is not only a 140 year old question that a few decades ago has been found to be unprovable from the standard axioms, but even since has there not been "good" axioms other than the claim itself that would decide it one way or another. Afaik.

So that out of the way, what's a notion to "believe" in CH that can be held without also believing in the constituents of it's formulation (e.g. at least aleph null, i.e. an infinity)?

>>11628349
>>11628475
Yeah? Well read pic related, you vapid cunt. I made a diagram clearly illustrating these high-level concepts to your feeble brain.
>Actually, if you don't believe in infinity then you MUST believe in CH
>you MUUUUUST
>NOOOO YOU CANT JUST DENY INFINITY AND CH!!!!
You are the most cock-hungry faggot retard on this board. I am surprised, you being so offensively stupid, that you even possess the ability to operate a computing device to post here.

>>11628469
I think you're being a predicativist
https://en.wikipedia.org/wiki/Impredicativity

Bro, CH is false, [math]2^{\aleph_0}=\aleph_2[/math], since Martin's maximum is true.

>>11628581
nah there's actually just a biggest number

>>11628600
It is true. I never get higher than 20 1/2, and only if I also take off my socks and my nappy.

>>11628581
haha, very based as the kids say, i love it
>>11628600
>>11628648
This is clearly false, you can always just keep adding...can always take a product...can always exponentiate....always exponentiating...subsets of subsets of subsets....powersets of powersets of powersets of powersets...on and on and on and on and
*dies*

>>11628133

it's such a boring and obvious result cloaked in esoteric symbols and proof methods that no one uses. it's the "numbers which we don't know the size of" equivalent of logic/meta-math. Wow there are numbers so big we don't know how many digits are in them? It's just stupid. Who gives a shit. It's a low-brow topic for morons to chatter about.

>>11628569

Lmfao this perfectly illustrates why set theory is so pozzed. People arguing about fucking nothing.

>>11628749
You're saying Godel's incompleteness theorems are obvious, both of them? What's your intuition?

>>11628862
He has been sniffing his own farts.

>>11628862
>it's the "numbers we don't know the size of" equivalent of logic/meta-math

>>11624308
>One implication is that math is not merely a mechanical game of formula-shuffling.

It is but I wish it weren't.

inb4 autism

ready """""""normal"""""" people?

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

>>11628894
I'm not really sure what that means and my interpretation of that phrase is nothing like Godel's theorems.

>>11628885
>buttblasted midwit who actually fell for the "understanding godel's theorems makes me smart"meme

So predictable.

>>11628909

Please, elaborate on how your nuanced interpretation of godel's theorems is so RADICALLY different from the fact that there are numbers we can't calculate the number of digits of.

You people really can't see the bigger picture can you?

>>11628919
>complicated result, with complicated proof, that stumped the greatest mathematicians at the turn of the century
>boring and obvious

>>11628927
>please elaborate on my analogy
It's your analogy anon. What's the similarity?

>>11628927
>there are numbers we can't calculate the number of digits of
Like what? Can't you just calculate floor(log10(n))+1?

>>11628940
>think of something obvious
>describe it in literally the most convoluted way you can think of
>profound result

Are people on /sci/ this naive?

>>11628961

Lmfao

>>11628959

I've already stated my analogy. You offered your interpretation as being different, but withheld it, I asked you to reveal your interpretation, despite that I revealed my analogy, and now you are avoiding sharing your interpretation. The only one hiding something here is you buddy.

>>11629093
>think of something obvious
I'm not the Empire of Dust guy, but I agree that it's tiresome. It's not clear if people like you are trolling or whether it may be worth the time explaining things.
Fact is, you would agree, that at least logicans and mathematicans overall didn't expect, in the 1890's, say, that there's well-formed formulas in arithmetic that, in Peano arithmetic, can neither be proven nor proven false.
So it's, my no good stretch of the word, "obvious."

Even if you personally may perceive yourself on an ontological high ground, the fact that most experts in the field for the longest time didn't expect it is a good way for a valid definition of "no obvious". Again, that's entirely independent of your personal conceptualization of the situation.

>>11629093
what's an example of a number that we can't calculate the number of digits of?

>>11628571
im not rejecting self reference, im just rejecting contradiction and assuming an object exists before even analyzing it (so i reject defining a contradictory object and being like, woah, paradox!)

>>11629118
a number that would require more symbols than will be generated in the lifetime of the universe

>>11629175
>assuming an object exists before even analyzing it
Yes, predicativism rejects that you can do this (iirc)

>>11629188
i see. is there any benefit to being a non-predicativist?

>>11629181
how do you know the universe has a lifetime?

>>11629192
You can prove more stuff, or maybe just prove it faster

>>11629198
This.

I mean advantage is relative but yeah.
E.g. consider this axiom:
>There's witches which can pull every meal out of a hat.
In more formal terms, think of the axiom of choice in stead of the witch sentence, maybe in the form
>Every surjective function has a right inverse

Now you can derive a lot of theorems from that. Example:
>Theorem 1. There's witches which can pull Kebap out of a hat.
Proof:
>Kebap is a meal. Using our axiom, Theorem is 1.

Another example:
>Theorem 2. There's witches which can pull Pizza out of a hat.
Proof:
>Kebap is a meal. Using our Pizza, Theorem is 2.

In that sense, the stronger the axioms, the more theorems you can prove.

>>11629198
This.

I mean advantage is relative but yeah.
E.g. consider this axiom:
>There's witches which can pull every meal out of a hat.
In more formal terms, think of the axiom of choice in stead of the witch sentence, maybe in the form
>Every surjective function has a right inverse

Now you can derive a lot of theorems from that. Example:
>Theorem 1. There's witches which can pull Kebap out of a hat.
Proof:
>Kebap is a meal. Using our axiom, Theorem is 1.

Another example:
>Theorem 2. There's witches which can pull Pizza out of a hat.
Proof:
>Pizza is a meal. Using our Pizza, Theorem is 2.

In that sense, the stronger the axioms, the more theorems you can prove.

>>11628569
Retarded faggot. State CH in the formal language of set theory. I will then show you how your own statement must be true if axiom of infinity is false.

>>11629093
You don't understand Gödels theorems nor their significance. If you think you can prove them in a simpler, less convoluted, way, please do. Clearly you cant because you fail to even understand why the results are not obvious (or are you calling Hilbert a retard?).

>>11628343
The continuum hypothesis is vacuously true if there are no infinite sets.

>>11629107
>>11629321

How is it tiresome? You are even admitting that you're essentially just trying to recreate the mindset of someone from over 100 years ago. It's a fucking LARP if you, someone who was born with a computer, who has had all the opportunity to speculate on what can and cannot be calculated from birth to adulthood, could not conceive of their being some wff that isn't provable. The analogy is right there. And besides that, inaccessibility as a general concept isn't even new. That's the part about this that's even more confusing. Surely there wasn't some kind of glitch in the matrix that caused the most intelligent men of their generation to suddenly be incapable of comprehending the idea inaccessibility? Surely they weren't actually naive enough to think that a finite mind could be all knowing?

My point is that godel's theorem being profound is hype, plain and simple. If you work through a logic I and II course, and you do it with a SOBER mindset, it's kind of like- huh that's neat. The only reason people think it's profound is because they're playing along. It's a social meme, a game. Recursion is interesting, functional programming is interesting, but it's really not as profound as people make it out to be. It's like a clever party trick. Who can be most blown away wins the most points. Like a said, it's boring, I'm not interested in it. There are far more interesting results in even baby number theory that I honestly can't give an immediate intuition for. I can pull a dozen inequalities out of my ass that are only undergrad level that you have to spend far more time giving an intuitive explanation of than the time it takes to find godel's thereoms intellectually agreeable.

>are you calling hilbert a retard

If you actually read math history (no one on /sci/ does), you will realize hilbert was not exactly the brightest of his generation.

>>11629310
>>11629333
Yeah okay.
If your statement was just about using the semantic artifact of conventions for predicate logic, you could have said that earlier.

"All witches know how to play Sega Sonic well"
being "true" because there's no witches and [math] \forall [/math] being making everything true on the empty domain is more an argument for Relevance logic than anything else. As in
https://en.wikipedia.org/wiki/Relevance_logic
I.e. shows that FOL doesn't capture reasoning well and could be upgraded.

>>11629343
I don't disagree with your general sentiment.
Although I disagree that it should have been evident to people 100 years ago, and that opinion is just informed by the actual history.
I haven't hyped the result, I've just given details to those who asked.

>>11629343
What do you consider to be interesting results in baby number theory?

>>11629310
>in the formal language of set theory.
This is why you're wrong, asshurt fudgepacker.
Nobody is obliged to use your retarded axiom system.

>>11629333
Depends on how you formulate it, you could say CH is true exactly when there exists a function f between the collection of all subsets of the smallest inductive set and the elements of the smallest uncountable ordinal

>>11629418
^bijection

>>11629418
If there are no infinite sets, your statement is also vacuously true. Formulate your statement in formal logic if youre still confused as to why.

>>11629444
>>11629418
To be a little bit more explicit, it's true because the empty set is such a function.

>>11629444
Just to be clear you are arguing that if a theory doesn't prove the existence of infinite sets then any statement that says "there exists a function on an infinite set..." is trivially true? This is not right, we could similarly just make the statement saying there is an injection from omega_2 into P(N), then CH would fail. In general universally quantified statements are vacuous but not existential statements.

>>11629545
No I said if you assume there are no infinite sets, not if you dont assume there are any.

>>11629545
To formalize your statement you have to universally quantify to define the smallest inductive set. Once you do, since there arent any, your statement will hold true.

>>11629192
Isn't non-predicativism inconsistent because of for example Russell's Paradox?

>>11629549
You are telling me the truth value of CH is true in V_(omega). I doubt this, do you have a citation?

>>11629561
Thats not what I said. Can you prove there are no infinite sets in V_omega? If you can, or if you assume there are no infinite sets, then yes CH is provably true.

>>11629395
He didn't mention any specific axiomatic system, just a set theory in general. If you refute the concept of formalization you cannot be argued with.

>>11629568
Yes, V_omega satisfies the negation of axiom of infinity. It proves that every set injects into some natural number. Please provide a citation or give a full proof.

>>11629577
It does satisfy it but whether or not you can prove negation of infinity inside it depends on what axioms you take. If you take ZF -infinity then you cant prove either that there are no infinite sets nor CH(by Godels theorem).
> It proves that every set injects into some natural number
Under what axiomatic system?

>>11624275
Sound like Godel was pretty retarded.

"This statement cannot be proved" can be proved wrong.

>>11629587
Listen you need to either prove your statement or provide a citation or I'm just calling bullshit. V_{omega} is a model for ZF with axiom of infinity replaced with its negation. When I said it proves, I meant it satisfies.

Why is it so hard for highschoolers to understand the concept of something being vacuously true?

>>11624293
prove it

>>11629602
I dont need to give you a paper, I can prove it here and now, assuming the negation of axiom of infinity. Give me a formal statement of CH you would like me to prove and Ill do it.

>>11629611
Burden of proof is on you bud, you are the one making claims. Still waiting

>>11629602
>>11629611
>>11629628

If you dont want to state CH yourself, I can do it for you:
"for all x,y,z (x is the set of natural numbers and y is the set of real numbers and z is a subset of y) => (|x|=|z| or |z|=|y|)"
Are you satisfied with this (almost) formal statement of CH?

>>11629643
I keep hounding the dude for a proof because he is wrong, thats why he can't come up with a proof. The point is that CH proves infinite sets exist, just look at the statement. It says there exists a (bijective) function from \omega_1 to the set P(N), and P(N) is provably infinite. There can be no model of ZF with the negation of the axiom of infinity that satisfies CH.

>>11629418
Nice formulation. For the sake of being rudimentary, I'd even pin omega 1 down more by calling it the set of all countable ordinals.

>>11629643
At the very least z must be restricted to be infinite.

But I'm a bit skeptical about this statement, since what you wrote is not about subsets of P(N) smaller than P(N) not overshooting N, whereas CH is usually stated about P(N) not overshooting the smallest uncountable infinity (or |\omega_1| if we're using ordinals).

I have no stake in your guys quirrel, but let me do some work for you guys.
We may write
[math] |\omega_0 \to \{0,1\}| = |\omega_1| [/math]
as
[math] \forall w. \ (\ P(w) \to |w \to \{0,1\}| = |\omega_1| \ ) [/math]
with P(w) denoting, I think,
[math] \forall N. \ ( \left[ \forall n. (n\in N leftrightarrow P_{ \omega_0 }(x) \right] \to |x \to \{0,1\}| = |\omega_1| [/math]
where the predicate is the [math] \leftrightarrow [/math]

>>11629418
Nice formulation. For the sake of being rudimentary, I'd even pin omega 1 down more by calling it the set of all countable ordinals.

>>11629643
At the very least z must be restricted to be infinite.

But I'm a bit skeptical about this statement, since what you wrote is about subsets of P(N) smaller than P(N) not overshooting N, whereas CH is usually stated about P(N) not overshooting the smallest uncountable infinity (or |\omega_1| if we're using ordinals).

I have no stake in your guys quirrel, but let me do some work for you guys.
We may write
[math] |\omega_0 \to \{0,1\}| = |\omega_1| [/math]
as
[math] \forall w. \ (\ P(w) \to |w \to \{0,1\}| = |\omega_1| \ ) [/math]
with P(w) denoting, I think,
[math] \forall N. \ ( \left[ \forall n. (n\in N \leftrightarrow P_{ \omega_0 }(x) \right] \to |x \to \{0,1\}| = |\omega_1| [/math]

>>11629665
Youre wrong. Youre confused because you dont actually know what CH asserts, formally speaking. It does not assert the existence of infinite sets. Again, try to state CH in the language of set theory and youll see what I mean.

>>11629418
Nice formulation. For the sake of being rudimentary, I'd even pin omega 1 down more by calling it the set of all countable ordinals.

>>11629643
At the very least z must be restricted to be infinite.

But I'm a bit skeptical about this statement, since what you wrote is about subsets of P(N) smaller than P(N) not overshooting N, whereas CH is usually stated about P(N) not overshooting the smallest uncountable infinity (or |\omega_1| if we're using ordinals).

I have no stake in your guys quirrel, but let me do some work for you guys.
We may write
[math] |\omega_0 \to \{0,1\}| = |\omega_1| [/math]
as
[math] \forall w. \ (\ P(w) \to |w \to \{0,1\}| = |\omega_1| \ ) [/math]
with P(w) denoting, I think,
[math] \forall n.\ (n \in w \iff ([n = \emptyset \,\,\lor\,\, \exists k. ( n = k \cup \{k\} )] \,\,\land\,\, \forall m \in n. [m = \emptyset \,\,\lor\,\, \exists k \in n. ( m = k \cup \{k\} )])) [/math]

(edit: corrected)

>>11629681
Youre right, but the issue is that w_1 is not a thing in the language of set theory. You only have the variables, €, logical connectives and quantifiers. You need to define w_1 in your statement. As soon as you do, your statement becomes vacuously true.

>>11629692
Write out your statement of CH in one sentence in full.

>>11629685
I'll try this one more time, assume CH, then there exists a bijection f between \omega_1 and P(N). The function f as a set or ordered pairs is infinite, since omega_1 is infinite and P(N) is infinite. Hence infinite sets exists. So CH proves that infinite sets exist. Therefore there can be no models of CH where there are no infinite sets don't exist. The only models of ZF with the negation of the axiom of infinity are those where all elements are finite.

>>11629692
On second read, its provable that for all x, not P(x) with your definition of P, if you assume there are no infinite sets ( thats equivalent to the assumption). So your statement of CH is vacuously true.

>>11629716
Dumb and wrong. As I said, state CH in the language of set theory to see why its true if you assume there are no infinite sets. To state you cant use the terms omega_1 and P(n), because theyre not terms in the language of set theory.

>>11629716
Youre retarded. Read my posts. Why do you keep saying the same thing without responding to my objections. Get a brain, moron.

>>11629730
Pure brainlet.
>>11629739
Point out the error then.

>>11629716
>>11629730
I can't tell who's who but best not use my P(n) as a set, I don't know why
>>11629716
says
> bijection f between \omega_1 and P(N)
or why
>>11629730
says P isn't in the language of set theory.
Both of your claims seem wrong in that instances

>>11629704
Well I'm not going to expands the definition of \omega_1 for you guys.

But let me see how far I can get with your or his (who ever of you is the "vacuuously true guy" Shyamalan rhetoric guys) claim.
I suppose it's >>11629720

Say w, r are two set variables and Ind(w) and U(r) say that w is the smallest inductive set and r is the first uncountable sets.

The idea, I understand, is to recast CH as

forall r, w. U(r) => [Ind(w) => ch(w, r)]

where little "ch" means
|w -> {0,1}| = |r|
where "=" is the existence claim of a bijection

So dissolving the implications under the forall as
P=>Q ... ]not P] or Q,
then CH is

forall r, w. [not U(r)] or [[not Ind(w)] or ch(w, r)]

i.e.

forall r, w. [not U(r)] or [not Ind(w)] or ch(w, r)]

and if we take the second disjunct as an axiom, we should be done.

>>11624275
Imagine a world where you have the real numbers, addition is a group, multiplication without 0 on the positives is a group, but you don't yet know you have a field. Can you prove, "the unique square root of 4 is 2?" No. and in fact, you can't disprove it without using language that would imply you have a field, or just plain old guessing, because that involves testing infinitely many negative numbers on the interval (-1, 0). Thus, this question is undecidable, and you would be considered the next promising young genius mathematician for "working backwards" by realizing that by using the commutative property of multiplication, you can treat the square of negatives like positives. That's one of Godel's theorems in a nutshell (I forget which one).

>>11629748
I wont engage with you anymore, retard. Youre like a child. Ive pointed out your mistake numerous times yet you refuse to listen. Kill yourself.

>>11629754
done with simplifying anyway, tell me if that kind of formulation is what you wanted to get at or where the misinterpretation is
?

sorry for reposting this, if you replied to it in the meantime. I don't want to have people more confused with half-finished sentences

>>11629716
>>11629730
I can't tell who's who but best not use my P(n) as a set, I don't know why
>>11629716
says
> bijection f between \omega_1 and P(N)
or why
>>11629730
says P isn't in the language of set theory.
Both of your claims seem wrong in that instances

>>11629704
Well I'm not going to expands the definition of \omega_1 for you guys.

But let me see how far I can get with your or his (who ever of you is the "vacuuously true guy" Shyamalan rhetoric guys) claim.
I suppose it's >>11629720

Say w, r are two set variables and Ind(w) and U(r) say that w is the smallest inductive set and r is the first uncountable sets.

The idea, I understand, is to recast CH as

forall r, w. U(r) => [Ind(w) => ch(w, r)]

where little "ch" means
|w -> {0,1}| = |r|
where "=" is the existence claim of a bijection

So dissolving the implications under the forall as
P=>Q ... ]not P] or Q,
then CH is

forall r, w. [not U(r)] or [[not Ind(w)] or ch(w, r)]

i.e.

forall r, w. [not U(r)] or [not Ind(w)] or ch(w, r)]

and if we take the second disjunct as an axiom, we should be done.

done with simplifying anyway, tell me if that kind of formulation is what you wanted to get at or where the misinterpretation is
?

>>11629754
Yes, you understood why CH is vacuously true if you assume the negation of infinity. Congratulations.

>>11629765
Now can you think of a formulation of CH where assuming the negation of axiom of infinity it's NOT vacuously true? You can't, because there is none. All formulations of CH are true if there are no infinite sets. That other guy is an absolute mong retard.

>>11624275
Yeah fuck Chegg. Pay for logic tutoring @ paypal.com/chillfill $40/hr and I won't sell you out if you wan't me to help you on an exam. If you just want a homework answer I can do that too for $10/problem. Don't get fucked by academia's bullshit
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>>11629760
Your objections make no sense, you just keep screeching vacuous without showing what is vacuous. Let's try and parse what you are saying, take a theory T in the language of set theory with the negation of the axiom of infinity. If you say that CH is true whenever the negation of the axiom of infinity is true, then in any model of T, CH holds. Then by the completeness theorem T must prove CH. But this is absurd since in the presence of CH infinite sets always exist, since the function that witnesses CH holding is itself infinite. You are wrong, be humble and admit it.

>>11629846
The function need not be infinite and actually for CH to hold the function neednt even exist, because in the statement of CH you preface it all with universal quantifiers to define what you mean by N (the naturals) and R(the reals). Im utterly amazed how you can know about compactness theorem yet fail to grasp that N and R are not valid terms in the language of set theory.

>>11629846
>But this is absurd since in the presence of CH infinite sets always exist, since the function that witnesses CH holding is itself infinite.

>in the presence of CH
What is the "presence" here?
Consider the statement S that N is in bijection with the ordinal 7. Does this statement S, that's about an infinite set, induce a "presence"?

>since the function that witnesses CH holding is itself infinite.
That sounds like a metalogical statement and I'm not sure if it fits into the theory/model you speak of itself.

I'm not Shyamalan btw.

>>11629887
Why are you referring to me as Shyamalan?

>>11629890
>>11629364

>>11629896
Its not a twist at all though, its only a twist for someone unfamiliar with basic logic and the notion of being vacuously true. What would be a twist is if CH actually implied the existence of infinite sets!

>>11628289
Weak

>>11629887
In the presence of CH I mean that CH holds in whatever model you are looking at. CH is the statement of the existence of a bijective functional relation between omega_1 and P(N). Even if omega_1 and P(N) are not the real omega_1 and P(N) of V any other model will agree on their infinitude this since there are injective functions f:\omega \to \omega_1 and f:\omega \to P(N) are in V. Since omega is absolute every model will agree these sets are infinite.
>>11629883
I'm not sure what you mean valid terms. They are names for formulas that are way to cumbersome to write down in the language of just membership. I know this is going to surprise you but I do know a lot of set theory.

>>11629952
>They are names for formulas that are way to cumbersome to write down
Why dont you try anyway?

>>11629739
Nobody knows which posts are yours, anonymous retard. You have to go back now.

>>11629997
>>11629984
>>11629952
>>11629916
>>11629910
Heh, guys, why don't you settle this the smart way and ask an expert? I have expertise in computability, mathematical logic, and set theory. 10% off if you buy in 15 minutes!

>>11624293
1+1=2
1+1=1
1+1=2+n
it's a question of axioms and what you want to use numbers for.

always amusing to see people desprately try to obfuscate and play down the meaning of Godel's theorems
>it's not surprising at all!
nice cope, faggot

Does this mean end of math?

Ok call me a faggot but I’m going to try and turn these sentences into logic please correct me or whatever

“This statement cannot be proved”
If statement = true
then statement = false
If statement = false
then statement = false

“This statement is a lie”
If statement = true
then statement = false
If statement = false
then statement = true

it gets into a loop in both situations, the first one loops by saying false = false and the other one loops by saying true = false or false = true

>>11631404
That's good, but this was known since the beginning of time. The whole point of Gödel's theorems is that the language is much more formal and made to be able to talk only about natural numbers. People thought there were no way to make it talk about itself to create such a paradox because it was a very strict language talking strictly about natural numbers. The ingenuity of the theorems is that Gödel was able to construct a statement strictly about natural numbers that was actually talking about itself.

>>11624275
It is actually proof that the logic thing has limits.

>>11631404
More on your sentence, note that in elementary arithmetic,
"this statement is a lie" is fundamentally different from "the statement cannot be proved" . Statements can be encoded in elementary arithmetic and provability can be encoded in elementary arithmetic so the "“This statement cannot be proved” can be expressed in elementary arithmetic, which is precisely Gödel's theorem, but "This statement is a lie" cannot be expressed in elementary arithmetic, because you cannot express truth/falsehood in elementary arithmetic. There is no way to say that some statement is true or false in elementary arithmetic, and the proof (due to Tarski) is by contradiction precisely using your statement.

>>11629575
>>11629310
>hurdur there's no axiom system, just use sets and actualize infinity heh
Retards. You're made for each other.
CH depends on the existence of the reals, which can be denied.

>>11629665
>The point is that CH proves infinite sets exist
No it does not, you absolute imbecile.
This is the dumbest shit I've seen on /sci/ in years.

CH is of the form
[math]\forall x\forall y \forall z (P(y) \wedge Q(x)) \implies CH(x,y,z)[/math]
Where P(y) is the formula that y is the set of natural numbers and Q(z) is the formula that asserts z is the set of real numbers.
If P(y) is not true for any y, CH holds.

>>11632699
CH is which formula?

>>11632735
[math]CH(x, \mathbb{R},\mathbb{N}) := x\subset \mathbb{R} \wedge (\neg finite(x)) \implies \exists f, dom(f)=\mathbb{N} \wedge range(f)=x \wedge bijection(f)[/math]

>>11631494
It's not so much that logic has a limit, but that our current axioms have limits to what they're able to prove, incompleteness theorem style. Theoretically, if we could add another axiom to ZFC with CH as a consequence, while keeping it consistent, there wouldn't be a problem, but nobody has really found any axiom we could add which does that.

I've always wondered if the implications of the completeness theorem were overstated or not, but it was always too boring to get into the details. But, like how we've proven you can't do certain things with just a compass and a straight edge, but you can with other means. It seemed dubious that the way Godel purportedly did things is an ultimate proof of unprovability.

>>11629883
This.

>>11628216
is this just saying that liar sentences have no truth value? it better not be because this was known since ancient fucking times