>tfw I took 20 mins to get my head around pic related. Anyone

>>7711952

Post feels relevant to having a low IQ but high interest.

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So R is the set with the elements x that are not in x. But x = x, so the elements x that are not in x is nothing, which would mean R is the empty set. What's wrong with this reasoning?

>>7711955
x themselves are sets, not elements.

Thought most people would be familiar with pic related sorry, should have clarified.

R is the set of all sets "x" where x is not a member of itself.

>>7711955
>>7711962

Also, see "Russell's Paradox", by "getting my head around" this I meant just generally realizing why shit is going to shit.

i.e. internalizing pic related (including the lower bit).

If your were smarter OP you would have your instructor explain to you in a third of the time.

You probably didn't just immediately say "Either R is in R or R is not in R". If you ask that question and then consider both cases it takes about 5 seconds to wrap your head around it.

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>>7711950
reminder that there is nothing wrong with R not being in R.

reminder that russel paradox is only a paradox is you want to call it like this.

reminder that russel solution as type theory is nothing impressive for anybody not a degenerate rationalist.

>>7712038
>there is nothing wrong

there is everything wrong. a set can only be built as a subset of another one

>>7712050
>what are non-well-founded set theories

>>7712079
i'm intrigued

>>7712079
>the birth of a meme

great, just like we can't have proper analysis discussion the board because some idiot who doesnt even study math comes spouting "oh my bullshit is true in nonstandard analysis!" we're going to have it happen with set theory too

this is why we can't have nice things

>>7712100
You should check that salt as it's clearly related to OP's post

>>7712104
The didactic, proper answer to it is >>7712050
Appealing to obscure theories with reduced application and specific topics isn't nice

>>7711950
lolwut its so simple

>>7712105
But that post it was replying to was about type theory, what I should have said is "Yes but that is not necessarily the case in type theory that was evoked in the post you replied to".
I'm not >>7712038 in case thought so.

>>7711950
Can someone more adept at maths than me explain why this shit matters? I understand the fun in a lot of logical paradoxes, but this one seems so trivial to me. Logical paradoxes are supposed to be intriguing, this is like Jaden Smith tier shit.

>how can object R be a set of all sets where the object is not a member of its own set, when i arbitrarily contradict myself with "then" statements?

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>>7712241
>forgot pic

>>7712241
>Logical paradoxes are supposed to be intriguing

No, they are not.

This shit matters because it is a false statement deduced from a set of axioms,
thus you can prove anything and its negation as soon as you introduce the comprehension principle. This is a serious problem, as math is about proofs.

>>7711950
kek I feel ya OP. The forward implication I recognized immediately, the reverse took me a few minutes. What's sad is I've seen this before and made sense of why it is true, and then a year later it still took this much time

>>7711950
Set theory is a [math]meme[/math] theory

>>7712970
That was an unnecessarily hurtful thing to say

>>7711950
>2015
>being retarded

>>7712241
It's a logical inconsistency arisen only from axiomatic properties. It could imply that proofs constructed from these axioms are not in fact valid. Set theory has become a pretty important branch of math. linear algebra, statistics/probability, and discrete math are examples of math that incorporate set theory.

Doesn't this just mean such a set doesn't exist? Math men said that about the answer to 1/0, they didn't throw their hands up in the air in confusion

>>7713190

model theory is GOAT. Weak-ass set theory btfo by Tarski

Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves.

>>7711950
i feel like you just posted something with terrible notation and that is the only issue

shouldnt these x's be of different cases (uppercase/lowercase) to differentiate them

>>7713251
no, they're the same x

>>7713251
>sets can't be elements

[math]x \in x[/math] means a set which contains itself

>>7713251
it's confusing, but not terrible.

it just means "the set of sets that don't contain themselves"

>>7713262

Lol

>>7713207
How is it used in theoretical computer science? Or is it all that much?

>>7713319
We use it in theoretical CS as a tool to describe matters. Like automatons and formal languages and so on. To define what's an automaton A = {states, input, start-state, accepted-states, value-function} is often used.
Makes it easier to talk about matters

>>7713270
The confusing bit is:

Does R belong to itself or does R not belong to itself?

Rest is filler.

>>7713825
well you can definitely see it's both, just by the definition of the set. if you understand the definition then that double implication is pretty clear.

>>7713319
I was going to mention CS but I chose discrete math instead, but like >>7713366 said, with languages and context-free grammars you'll need set notation. Structural induction is a another biggie that needs sets. Any proof for completeness of connectives (and, or, not, etc) needs structural induction to prove. Set theory has worked it's way into a lot of stuff.

There is no such set which contains itself. Therefore R is a set which contains all sets. This not proof, just contradiction OP. Whoever made that pic (not you OP) is either genius or a real funny prankster. Breaking the set axioms doesn't mean you're genius, it just means you fucked up

>>7713964
Russell's paradox nigger, plus a set is any collection of objects (including sets). Given the axioms of set theory, this is a valid set.

>>7711950
>does a set of all sets contain itself
Sounds meme

>>7713988
Not really, its the set of sets where the set is not a member of its own collection of sets, and either the original set is a member in itself (which would violate the definition) or the set is a set that does not contain itself.

>>7713964
>Breaking the set axioms doesn't mean you're genius, it just means you fucked up

as if everybody sanctifies those axioms like you

>>7713231
No. You do not asume that it exists and show a contradiction. By the comprehension principle it does exist.

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interdasting

http://www.nature.com/nature/journal/v528/n7581/full/nature16059.html

>>7713319
>>7713944
>>7713366

>ctrl + f "complexity theory"
>0 results

What's wrong with you niggas?

>>7714082
are you this guy ?
http://mathoverflow.net/users/22002/mozibur-ullah

Is x an element or a set? Is R a relation or the real numbers? Fucking use the correct notation.

>>7711962
Wait, so x is a set of sets?

>>7715533
Since R is a collection of sets, it can be thought of as both.

>Russell
Into the shit can it went
So you believe anyone who is said to be smart Must be smart? That is your paradox right there

>>7715547
But x is not noted as a set, it should be X, not x.

>>7715646
Also, given that x is a set, the use of the contain symbol is not correct.

>>7715646
In set theory there are only sets.
There just happen to be sets that are elements of other sets.

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Who high IQ low interest here? I don't know what that funny E symbol means but I don't have to because I have a high IQ and everything in life comes easy to me.

>>7715646
>>7715659
you don't understand set theory

>>7713240
>that "aahhhhh-ha" moment after reading this

thanks bro

>>7715691
I mean, just looking at all this, I guess it means "contains" like [5,1,4] E 5

>>7715759
Shit I was actually right when I was joking around. Darn Python teaching me about lists.

>>7711950
>Naive set theory
LOL
Let's work in NBG set theory.

Let [math]M(X)[/math] be a property of [math]X[/math] being a set. That is, [math]M(X)[/math] is defined as [math](\exists Y)(X \in Y)[/math]

We can define a class R: [math] R = \{ x | x \notin x\}[/math] where [math]x[/math] is a set.

It is obvious that [math](\forall x)(x \in R \iff x \notin x)[/math] which can be written as [math](\forall X)(M(X) \implies (X \in R \iff X \notin X))[/math]

Let us assume that [math]M(R)[/math]. Then [math]R \in R \iff R \notin R[/math] which obviously leads to contradiction.

Therefore [math]\vdash \neg M(R)[/math]
[math]R[/math] is a proper class.
There, no paradox here.

>>7713964
>There is no such set which contains itself
[math]x = \{x\}[/math]
Do you even LISP.

>>7715762
>We can define a class R: R={x|x∉x} where x is a set.
>x is a set that has no elements in itself.

How is this not just an empty set [math]\emptyset[/math] anyway?

So nothing is an element of nothing if and only if it's an element of nothing.

I don't understand the point of set theory. Have mathematicians ever actually got something useful out of it?

>>7715779
>x is a set that has no elements in itself.
No. x is a set which does not contain itself. It can contain everything it wants, just not itself.
Sets can contain themselves, nobody forbade them to.

>Have mathematicians ever actually got something useful out of it?
Read the thread. A few of the posts discussed this already.

>>7714082
Emma, is that you?

>>7712245
This made be vomit

>>7713305
That's what it means, though.

>>7711950
R is a set whose elements do not contain themselves. Thus R belongs to R is equivalent to R does not contain R, since it belongs to R

>>7713305
He's right you straight up idiot

>>7715739
It's just a theory, so it's not real.

>>7716677
>maths are not real

this is what continental philosophers believe.

>>7716890
But our eyes aren't even real, then how can maths be real? (notice Maths and Mirrors both start with the same letter therefore both belong to the same group)
QUED

Do you have an example of a set that contains itself ? Can't figure how to write it

>>7717967
you can't in well-founded set theory, or there would be an infinite chain of elementhood. the point is to show a problem with an unrestricted LANGUAGE for describing formal systems axiomatically. Technically, every symbol in the formula is being used appropriately (sets can be elements of other sets, we can form predicates of sets by membership), yet when used in this way it leads to a contradiction; there is an element which is both in and not in the extension of the predicate.

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>>7718075
>you can't in well-founded set theory,
well-founded just means ''exclude at least russel paradox''

>>7712241
People tried to find a set of formal axioms that describe set theory and let us do math. Contradictions in a set of axioms can be really bad, in this case it meant that essentially you could prove any statement true. In layman's terms:
>Suppose the statement is false
>Invoke the axioms and work your way up to the contradiction.
>Since we have reached a contradiction, then our assumption must be wrong and the statement must be true.

In order to get rid of this paradox people patched set theory by adding in an extra axiom that says all sets constructed in this way must be subsets of another set. Therefore the set defined in OP is not legit set in modern set theory.

>>7713251
>>7715533
>>7715646
>>7715659
This flavor of set theory is defined in first order logic. Everything is a set in this theory.

This doesn't mean that you can't talk about things like numbers, n-tuples, or functions, but it does mean that you have to do a wealth of boilerplate math in order to "encode" those objects as sets.

>>7713366
To add to what other anons have said, it comes up in computability theory as well. One may talk about recursively enumerable sets and their relationship to computable functions and decidable/recognizable problems.

However, I believe that CS is beginning to move more towards type theory and category theory, which would be good in my opinion since they're better suited to the type of math people do in CS.

I also get the feeling that CS in general works in Von Neumann- Godel-Bernays (NBG) set theory where you have a new object called a class. This lets you resolve paradoxes because the class of all sets is not a set.

>>7715758
>that "aahhhhh-ha" moment
https://www.youtube.com/watch?v=cyIIYerdlUA

>>7715762
My nigga.

>>7711950
Yeah you have to have R be a proper class man.

>>7711962
>sets no uppercase
just fuck my notation up senpai

>>7719556
The thing to keep in mind is that everything in set theory is a set. If you made your sets uppercase then literally everything would be uppercase.

>>7719309
no dur but poster was asking for an example of one such set which if you are thinking intuitively about the set theories most people know (which are p much always well founded) then you will never bump into one.

otherwise, >>7717967, if you don't mind leaving your usual set theories, just stipulate that there is a set X whose only element is X (and so forth forever). but then you will have a non well founded set.

>>7719547
>in this case it meant that essentially you could prove any statement true.
not true.

the adoption of the principle of explosion remains a personal choice.


>>7719942
>if you are thinking intuitively about the set theories most people know (which are p much always well founded) then you will never bump into one.

I love how rationalist try to get out of their mess.

you reject axioms which have consequences that you do not like, at least admit this.

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>>7711950

>Nigger doesn't into Russell's Paradox
>mfw I'm talking to a bunch of words on a screen
>mfw I realize anon is probably out getting high

>>7719960
The principle of explosion is not like the axiom of choice, where you can decide to have it as an axiom or not. Rather the principle of explosion is part of the proof system for your logic.

So, yes it's a personal choice, but only in the sense that the logic you use is a personal choice. Classical logic contains the principle of explosion.

>>7720783
>>So, yes it's a personal choice, but only in the sense that the logic you use is a personal choice.
of course and a contradiction matters only in the logic level, not on the mathematical side.

>>7721359
Mainstream mathematics are all formalized on classical logic. There are other branches of mathematics that are formalized on different logics. Still the contradiction always matters, again it's not like the axiom of choice, we're talking about the proof system itself.

Literally the only time you can ignore formalism and foundations is when you're specifically talking about a model with no regard to an axiomatic system. I wouldn't call that "mathematics" anymore, rather it would be something more analogous to science or engineering.

Physics grad student here.
Damn, math notation is always so weird to me with sets and group theory.
Should I take a group theory course at this point in my career or just crack open a book on Lie algebra?

>>7721384
learn linear algebra fro qm and gr
+ group theory for qm and gr
plus diff geo for gm

>>7715691
it means "element of"
A = {6,2,4,5}
6 ∈ A
10 ∉ A

its not used much outside of theoretical math because of how complex it can get

>>7715691
You're gonna have one hell of an existential crisis when you realise you're actually not very intelligent.

>>7715091
No, but he's arguably asking the most interesting solvable questions on Physics and Philosophy SE.

>>7723929
>solvable
best meme

>>7711950
I'm probably decent IQ, high interest, but more or less made of problems and broken at best.

I've lost even the drive to improve anything, try anything new, or meaningfully change my life situation. I just want to slowly sink until I magically cease to exist.

I wasn't good enough.

>>7724237
>I wasn't good enough.

Ayyye don't be hard on yourself.

If you found out you had cancer today, you'd probably start realizing how much happier you could have been if you aren't fretting over things.

Pros of being a hypercondriac, you realize how sad you can be if this underappreciated ride is ending.

ok

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>>7711950
it took you 20 minutes to understand because every line in between has been removed, and they went straight to the conclussion

let R be a group that contains all groups that do not contain themselves
then R is a group that does not contain itself
Therefore R needs to be contained in R
but then R contains all groups that do not contain themselves, plus it contains R that DOES contain itself. So R is no longer R.
Thus R cannot include R.
The paradox can be solved if we remove R from within itself, but then the proof is incomplete, because we proved that R can exist as long as it does not contain itself.
The paradox exists >> cannot be proven
Thus, if the paradox does not exist, it can be proven

So the paradox does not exist

QED

>>7719547
>This doesn't mean that you can't talk about things like numbers, n-tuples, or functions, but it does mean that you have to do a wealth of boilerplate math in order to "encode" those objects as sets.

I saw another anon mention even numbers are encoded as sets but thought it must have been a typo. I was aware a function can be defined as a set (a set of 2-there's), and even that turned can be defined as sets. For example, (a,b) would be equivalent to { {a}, {a,b} }. But how are numbers assigned to sets? My guess is maybe let the empty encode 0, then maybe the set containing the empty set as 1, an so on?

>>7728620
Oops, "2-theres" should be 2-tuples, and "turned" should be tuples

>>7728625

Just testing mathjax:

$$ \int_{0}^{+\infty}f(x)dx $$

>>7728620
You say that there is a set [math]N[/math] such that [math] \varnothing \in N[/math] and [math]\forall n \in N \colon n \cup \{n\} \in N[/math].
This is not unique, but you can take the intersection of subsets that also fullfill this poperties and you get a unique set.

Now you say that [math]\varnothing = 0[/math] and [math] \forall n \in N \colon n + 1 = n \cup \{n\}[/math], e.g. [math]3 = \{0, 1, 2\}[/math].
Via recursion and the +1 operation you can define the usual addition and multiplication. With the naturals you can go on and construct the rest of the number systems.

Your idea of [math]n + 1 = \{n\}[/math] also works, but is less nice because you don't end up with ordinals.

>>7727589
All of those are implied by the logical operators there. You just don't understand formalized logical notation.

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>>7728620
in set theory, everything is a set. numbers are sets in set theory, because the sperglords having faith in set theory think that set theory provides a foundations for mathematics, aka other branches of maths which are not explicitly set theory.