/stg/ - set theory general

[Anonymous No.11512962]

[Anonymous No.11512971]

Is there really enough interest in set theory on this board to warrant a general? At least say something about it yourself :D

[Anonymous No.11512989]

Redpill me on continuum hypothesis. I know it's independent of ZFC but surely we know more about it now. I heard Woodin has interesting things to say about it?

[Anonymous No.11512999]

>>11512971
There isn't. The interest in set theory on this board is next to zero.
>>11512989
>2 posters
Nigga did you really make this thread for this?

[Anonymous No.11513108]

>>11512999
Yeah, we all know /sci/ is filled to the brim with more interesting IQ/COVID/0.999.../flat earth/prove me wrong/failed undergrad threads. Definitely no room for anything else.

[Anonymous No.11513111]

>>11513108
There really isn't room for a general based on a niche subtopic of an already existing not-that-fast general.

[Anonymous No.11513118]

>>11513111
A 1-post-a-day niche general is worth 20 of those worthless threads. Yes, a thread died for this... and it was 100% justified.

[Anonymous No.11513121]

Set theory is important because it is a theory of integers, models of axiom systems, infinite ordinals, and real numbers, all in one unified structure. This allows it to serve as a foundation for all of mathematics, anything you talk about in mathematics can be formalized in set theory naturally and easily, and studying set theory allows you to prove theorems about mathematics itself. The formulation of set theory in the late 19th century motivated the metamathematics of the 20th century, with all the astonishing results about provability.

[Anonymous No.11513123]

>>11513118
>A 1-post-a-day niche general is worth 20 of those worthless threads.
If that post was discussion, and not you endlessly necrobumping, then maybe, but we both know that's not how it's going to work.

[Anonymous No.11513126]

>>11513123
I’m only bumping because I’m replying to your inane posts, you twat. Stop making useless off-topic replies.

[Anonymous No.11513234]

Set Theory is kinda gay tbqhh

[Anonymous No.11513269]

>>11513234
You only say that because you know no set theory.

[Anonymous No.11513310]

>>11512989
The continuum hypothesis says that the size of the real numbers (the continuum) is strictly larger than the size of the natural numbers

[Anonymous No.11513323]

>>11513310
not quite on the mark

[Anonymous No.11513351]

>>11513310
Ah yes, the good old "Answer your own question incorrectly to bait people into correcting you." Gotta love it.

[Anonymous No.11513378]

>>11513310
No, that’s Cantor’s theorem.

[Anonymous No.11513381]

hey
infinite set

[Anonymous No.11513493]

>>11512962
Virgin set theory vs chad model theory

[Anonymous No.11513587]

>>11512962
What part is the bottom?

[Anonymous No.11513617]

Why yes I do assume the existence of [math]\omega[/math] many Woodin cardinals with a measurable above them all.

[Anonymous No.11513837]

>>11513587
Inaccessible?

[Anonymous No.11513888]

>>11512989
>independent on ZFC
>independent

nigga do you realize that's a euphemism for contradiction?

>>11512962

set theory is a fucking shaggy dog joke. sage

[Anonymous No.11513897]

>>11513888
bump

[Anonymous No.11513905]

>>11512989
There's a good description on https://plato.stanford.edu/entries/continuum-hypothesis/ but I don't know the most recent developments.

[Anonymous No.11514037]

>>11512989
>Redpill me on continuum hypothesis. I know it's independent of ZFC but surely we know more about it now. I heard Woodin has interesting things to say about it?
The continuum hypothesis (CH) is the assertion that any infinite set which is too big to be matched one-to-one with the integers has at least as many elements as the set of all real numbers.
The generalized continuum hypothesis (GCH) asserts that the smallest infinite set which is too big to be matched one-to-one to an infinite set S can be matched one-to-one with the set of all subsets of S.
The continuum hypothesis was formulated by Georg Cantor as a conjecture in his theory of ordinal and cardinal infinities. David Hilbert listed it as the first of his 23 problems for 20th century mathematics.
In 1940 Kurt Godel, extending earlier ideas of Hilbert, proved that it is consistent with standard set theory that GCH is true.
In 1963, Paul Cohen proved that it is also consistent with standard set theory to assume that CH is false. Cohen's method made it clear that CH and GCH are also unprovable in any extension of set theory which only adds axioms which assert the consistency of previous axioms.
Because the CH is independent of standard axiom systems, most mathematicians consider it an undecidable proposition with no definite truth value. Probabilistic intuition leads many to also view it as more false than true. Mathematicians with a more platonist view consider it an open question, to be settled by new axioms.

[Anonymous No.11514230]

>>11514037
>Cohen's method made it clear that CH and GCH are also unprovable in any extension of set theory which only adds axioms which assert the consistency of previous axioms.
Why? Something about forcing?

[Anonymous No.11514259]

>>11514230
You only have to show that no large cardinal axiom implies CH by showing that it is consistent with not CH. Let [math]\kappa[/math] be strongly inaccessble [math]M[/math], we show that [math]\kappa[/math] is strongly inaccessible in [math]M[G][/math], where [math]M[G][/math] is the forcing extension obtained by the standard Cohen forcing [math]P[/math] that makes [math]2^{\aleph_0}=\aleph_2[/math]. Let [math]\lambda<\kappa[/math] be given, then there are only [math](2^{\lambda})^{|P|}[/math] many names for subsets of [math]\lambda[/math] in [math]M[G][/math]. Since [math]\kappa[/math] is inaccessible then [math](2^{\lambda})^{|P|}<\kappa[/math], thus [math]\kappa[/math] is still inaccessible in [math]M[G][/math].

[Anonymous No.11514389]

>>11514259
So rank-into-rank [math]I_0[/math] is consistent with [math]\neg[/math]CH? I thought this was an open question.

[Anonymous No.11515115]

>>11514389
See Vitoria Gitman's answer.

https://mathoverflow.net/questions/155428/very-large-cardinal-axioms-and-continuum-hypothesis

[Anonymous No.11515544]

>>11512962
set theory is hella gay desu. this is now a type theory thread

[Anonymous No.11515914]

>>11515115
If I'm interpreting that answer correctly it shows [math]I_0[/math] is consistent with CH, but does it show it's consistent with [math]\neg[/math]CH? Sorry I'm a brainlet.

[Anonymous No.11515957]

>>11515914
Look at last sentence of her answer and plug in [math]\aleph_0[/math] for [math]\kappa[/math]. Basically she is saying that small forcings such as those fucking around with the continuum at low levels will not affect any large cardinals.

[Anonymous No.11517037]

Threadly reminder to work with Computabilty Theorists

[Anonymous No.11517796]

>>11517037
wtfw it's like smho tbfh

[Anonymous No.11518243]

How do you overcome Russell's paradox with von Neumann's construction of the ordinals?

[Anonymous No.11518268]

>>11518243
The two have nothing to do with each other.

[Anonymous No.11518273]

>>11517037
>Chong & Yu

[Anonymous No.11518275]

set theory or operator theory?

[Anonymous No.11518295]

>>11518268
How does the implication that [math] \mathbb{N}\in\mathbb{N} [/math] have nothing to do with Russell's paradox?

[Anonymous No.11518367]

>>11518295
Implication of what?

[Anonymous No.11518552]

>>11518367
>Implication of what?
Can you read?

[Anonymous No.11518560]

>>11518552
What are you, retarded?

[Anonymous No.11518563]

>>11518560
Answer the question or explain why [math] \mathbb{N} \not\in \mathbb{N} [/math] you fucking autist.

[Anonymous No.11518564]

>>11518552
Clearly state what you are thinking so we can correct you. The Von Neumann ordinals have nothing to do with Russell's paradox.

[Anonymous No.11518568]

>>11518564
I literally did. >>11518563

[Anonymous No.11518576]

>>11518563
>>11518568
[math]\mathbb{N} \not\in \mathbb{N}[/math] simply because [math]\mathbb{N}[/math] is not a natural number. Natural numbers are numbers like 1,2,3,...
Retard.

[Anonymous No.11518577]

>>11518563
that x is not. in x follows from

https://en.wikipedia.org/wiki/Axiom_of_regularity

[Anonymous No.11518579]

>>11518563
The ordinals are well-founded by definition, they have no infinite decreasing sequences. If [math]\mathbb{N}\in \mathbb{N}[/math], then there would be an infinite decreasing sequence in [math]\math{N}[/math]. Russell's paradox is different though, it is a contradiction. It shows that there is no universe of sets. If you want to understand why it is not the case that [math]x\notin x[/math] for any set [math]x[/math], look up the axiom of foundation.

[Anonymous No.11518581]

>>11518563
You've got to be one of the lowest IQ niggers on this board. God, you're so fucking retarded you don't even realize it.

[Anonymous No.11518586]

>>11518563
>>11518295
>>11518243

[Nameless No.11518591]

>>11518563
Surely all of the natural numbers are individually finite? And Von Neumann''s construction of natural number n is a particular set containing n elements? And the set of all natural numbers is infinite? Thus the set of all natural numbers is not one of the constructed natural numbers. Or were you thinking of something else?

[Anonymous No.11518594]

>>11518568
>>11518576
[math] \forall n\in\mathbb{N}, \{0,1,2,...,n-1\}\in n. [/math] Since every natural number is in [math] \mathbb{N}, [/math] how does this not imply that [math] \mathbb{N}? [/math]

[Anonymous No.11518596]

>>11518594
Because the set of natural numbers is not a natural number, idiot.

[Anonymous No.11518604]

>>11518594
Also
[math]\forall n\in\mathbb{N}, \{0,1,2,...,n-1\}\in n.[/math] this is wrong.

[Anonymous No.11518611]

>>11518604
Sorry, shouldn't have the braces.

[Anonymous No.11518615]

>>11518591
I guess this kind of hel

[Anonymous No.11518620]

>>11518591
I guess this kind of helps, but how can we just immediately say that once the cardinality is infinite, the element is no longer an element of the set? This doesn't appear in the definition, so I don't really understand how it follows.

[Anonymous No.11518625]

>>11518596
>Why isn't the set of natural numbers a natural number?
>Because it isn't, idiot.
Insightful.

[Anonymous No.11518627]

>>11512962
I recently did a survey of constructive set theory axioms (BCST, ECST, IKP, CZF, IZF, etc. etc.)

https://gist.github.com/Nikolaj-K/951d44b393a92acc51f915fcb0e31cc2

[Anonymous No.11518632]

>>11518620
Again, ordinals defined to be transitive sets well-ordered by the [math]\in[/math] relation. This means they can contain no infinite decreasing seqences. If [math]\alpha\in \alpha[/math] for some infinite ordinal [math]\alpha[/math], then [math]\cdots \in \alpha \in \alpha \in \alpha[/math] would be an infinite decreasing sequence in [math]\alpha[/math].

[Anonymous No.11518636]

>>11518625
>>11518620
Read, gorilla
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
>The set N of natural numbers is defined in this system as the smallest set containing 0 and closed under the successor function S defined by S(n) = n ∪ {n}.
If N was in N, then N+1 would be in N, meaning N+1<N, contradicting x+1>x for all ordinals x.

[Anonymous No.11518784]

>>11518243
>>11518295
>>11518563
ZFC does not have unrestricted comprehension, so you can’t assert that there exists a set that contains all sets that don’t contain themselves. You can, however, show that x in x is always false using the axiom of regularity/foundation.

[Anonymous No.11518798]

>>11518295
>>11518563
The other answers aren’t helpful. The helpful answer is that N is defined as the smallest inductive set (i.e. the intersection of all inductive sets).

An inductive set is a set that contains the empty set and is closed under ordinal-successor (i.e. if x is in the set then x U {x} must also be in the set).

The axiom of infinity asserts that at least one inductive set exists. You can extract N itself (in case this set is bigger than N) by using the axiom of specification/separation.

N not in N also follows more directly from the axiom of regularity.

[Anonymous No.11519454]

>>11512962
so what's your thread about OP?

[Anonymous No.11519583]

>>11512962

Is Set Theory a fucking psyop? Literally has ZERO application, at best it can only make OBVIOUS or trivial statements- there is constant squabbling and confusion among "experts," who can't agree on what axioms to use, you have people CASUALLY throwing around the word "independence" like it's not a massive red flag. Like what the fuck is up with set theory?

[Anonymous No.11519611]

>>11519583
Set theory is the path to discover another more fundamental true about the nature of mathematics

[Anonymous No.11519620]

>>11519611

pretentious garbage. give me the insights of Ramanujan any day over the hogwash of set theory.

Calculus? Awesome
Statistics? Fascinating.
Number Theory? Beautiful.
Geometry? Endless wonder.

Set theory? Trash.

Set theory isn't math. Set theory is mathematical fiction (at best) and should be reserved for discussions on /lit/ and /his/.

[Anonymous No.11519740]

>>11519583
>>11519620

[Anonymous No.11519757]

>>11519583
>applications

[Anonymous No.11519825]

>>11519583
>Is Set Theory a fucking psyop? Literally has ZERO application
Why would not having applications relate to being a psyop?
Science and math having to have "applicationa" is just capitalist subversion of a subject matter. Painting or a walk in the park doesn't have "applications" either, don't try to commodify ever human endevour.

>you have people CASUALLY throwing around the word "independence" like it's not a massive red flag
Sets in their naive form are natural. The fact that some statements about sets can't be captured via easy axioms isn't the "fault" of set theory, it's a feature of mathematical logic per se.

[Anonymous No.11520044]

>>11519740

Wow you're a clever one aren't you? Dumbass.

>>11519757
>replying with an image of a number theorist when I'm complaining about set theory

Also a dumbass. I'm not ONLY in favor of application it was just to make a point. I do math all the time for the pure joy of it. That's what I'm advocating for. My point is that set theory IS NOT math and thus it is pretentious cancer that rots the brains of young impressionable math students. You see the SAME exact thing in art school where people who are talented become impressed by "theories" of art and end up going from talented to making MFA trash. Or with engineering students who have a talent for building protoypes but become convinced they need to understand string theory instead. Utter trash. Set theory has the same effect. Ruins people who enjoy math for the sake of it and convinces them that it is somehow "superior" to "mere arithmetic" and whatever pretentious twaddle set theorists like to come up with. That's why I'm calling it a "psyop" which is obviously an imprecise accusation but the point remains, it is subverting impressionable students of math and steering them away from what could have been a fruitful career in doing actual math. I don't prefer applied mathematicians, many of them are too anti-intellectual for me- but at least they are actually doing math. Set theory is the emperor's new clothes of mathematics.

>>11519825

Commie scumbag fuck off and let me do my math in peace without ruining it with your shitty notation faggotry. I wouldn't be surprised if you were the type of person to "question" why someone would want to get better at mental math, or enjoy adding and multiplying numbers or playing with sequences and primes.

Critical theory ruined art.
String theory ruined engineers and physicists.
Set theory ruined math.

Theorists are parasites on the actual talent.

[Anonymous No.11520088]

>>11520044

[Anonymous No.11520116]

>>11520044
You can austistically screech all you want but set theory will still be math when you're done.
>Ruins people who enjoy math for the sake of it and convinces them that it is somehow "superior" to "mere arithmetic"
No one thinks this, what the hell are you talking about. I guarantee I know a lot more set theorists than you and no one has this holier than thou attitude towards the rest of mathematics that you think they do. Let me explain where I think this idea might be coming from. Many set theorists work high up in complexity hierarchies. A problem may be as simple as possible to them but complex to someone working doing something else. For instance someone working in descriptive set theory might say the isomorphism problem for FGA groups might be as simple as possible, but to a group theorist with specific groups in hand this might be a tough problem. He is not saying that the problem of that group theorist is trivial, he is saying that his techniques are not fined tuned for that type of problem.
>could have been a fruitful career in doing actual math
Again set theory is mathematics, what is mathematics to you? Many set theorists do have fruitful careers, some don't, many mathematicians have fruitful careers some don't. I'm not really sure what you are arguing here.
>let me do my math in peace
Then go bruh, no one said come into this thread. You just came in here started spouting your bullshit. Leave man.

[Anonymous No.11520126]

set theory is nonsense.

the set of all numbers does not exist, nor does it have a knowable size.

no set can be constructed to have infinite elements, even by mere assumption. It's quite literally an illegal nonsense process.

[Anonymous No.11520139]

>>11519620
Infinity's onoy definition exists because of set theory you fucking stupid gob nobbler. If set theory doesn't hold, infinity doesn't hold. Then calculcus and ramenpajeet doesn't hold, and most modern math comes tumbling down.

[Anonymous No.11520148]

>>11520139

literal kike bullshit.

[Anonymous No.11520158]

>>11520044
> it is subverting impressionable students of math
> a fruitful career in doing actual math.
> I don't prefer applied mathematicians, many of them are too anti-intellectual for me- but at least they are actually doing math.
Thanks to the internet you can actually throw you kiddy tantrum somewhere and shut the fuck up in real life where adults would slap your face when you start screeching like this.

[Anonymous No.11520202]

sorry /sci/ is a type theory board

[Anonymous No.11520206]

>>11520148
infinity is indeed kike bullshit

[Anonymous No.11520226]

>>11520126
The set of all numbers? What kind of numbers?

[Anonymous No.11520237]

>>11520226
All of them, retard.

[Anonymous No.11520255]

>>11520237
But there are many kinds of numbers. Be specific.

[Anonymous No.11520265]

>>11520255
I'm not him - it was a poor attempt at a joke.