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4:01 PM
Are you trying to understand it from Euclid's Elements?
 
@BalarkaSen hi
 
A year ago, as a plus he wanted to show us that infinities can be of different size, so he told us to compare all the numbers in the line of $y=x$ and all the lines in the $xy$ plane, and we all(him included) reached in an agreement that the plane has more points.
However, now that I learnt some set theory, $|\Bbb R \times \Bbb R|=(2^{\aleph_0})^2= 2^{2\aleph_0}=2^{\aleph_0}$
 
@UserX There's a bijection between the line and the plane, though
Indeed
$|\mathbb{R}|\sim|\mathbb{R}^n|$
 
@daOnlyBG I liked Stein and Shakarchi's 'Real analysis' when I was learning it.
 
And the points on $xy$ plane are an infinite subset of $R$, bigger than $\Bbb N$ though, so their cardinals are equal
 
4:09 PM
Anyone for this question ?
1
Q: Is this group finite?

RamanewbieLet $G$ be a sub-group of the invertible real matrices of size $n$, such that $\forall M\in G,M^2=I_n$ Is $G$ finite ?

@UserX No
 
So they have exactly the name numbers. Am I wrong somewhere? How can I formulate the set containing the points of $y=x$ in math notation?
 
@UserX They're a subset of $\mathbb{R}^2$, not of $\mathbb{R}$
 
Typo
Not typo in fact but yea, I know
 
@UserX $\{x\mid y=x\}$
That's the way to write the set
 
Oh... Thanks
Wait I got confused
 
4:13 PM
@TheGame That notation reads like $y$ is fixed; I assume UserX means the diagonal of the plane. So I would write it either as $\{(x,x) \in \Bbb R^2 : x \in \Bbb R\}$ or $\{(x,y) \in \Bbb R^2 : x=y\}$.
probably the latter
 
How do I know there is no cardinal between $\aleph_0$ and $2^{\aleph_0}$
 
@UserX It is undecidable.
 
@UserX Assume the continuum hypothesis.
 
You don't. That's known as the 'continuum hypothesis'.
@TheGame There are uncountably many such matrices even for $n=2$, so there are in general as well.
 
Hypothesis? So this is unproven but conjectured?
 
4:14 PM
@UserX no. It cannot be proven.
 
@MikeMiller $G$ is finite
 
Bu we don't know if it's correct either
?
 
@UserX You can assume that or the opposite of that. Both assumptions are consistent with ZFC.
 
@UserX Exactly so it is undecidable.
 
@DanielFischer Hello.
 
4:15 PM
It has been proven to be unprovable.
 
@TheGame I misread it asking about the size of the set of matrices that square to 1; demanding that it be a subgroup is a different situation entirely, you're right. Sorry!
 
So is there something wrong with the ZFC system?
 
@MikeMiller It is finite, but I do not now how to show it.
 
@TheGame Any such group, you mean?
 
@UserX No. Every axiomatic system will have results that cannot be proven
 
4:16 PM
@MikeMiller Right now I have two answers saying it is infinite :c which is false
@MikeMiller indeed
 
@Alizter What unprovable results does the Peano axiomatic system have?
 
@TheGame Jyrki gave a good answer.
 
@UserX Other than set-theory and a few parts the CH does not make any difference to most results.
 
So no need to worry about it
 
4:17 PM
@MikeMiller :)
 
However if you really want to it is generally assumed that there is no cardinal there.
 
If you don't read carefully (like I didn't) it's easy to read it as asking about the set of such matrices, which I suppose confused the other posters, too
Well, no, one of them just gave a completely useless hint.
 
@Alizter This is not true. There are theories that are both consistent and complete; these theories just don't satisfies the conditions of Godel's theorem.
 
@MikeMiller You take over I have read very little on this subject.
 
4:20 PM
well, just don't ask me to state the conditions of godel's theorems. :P
 
@Chris'ssis :P
This is seriously bad
How comes two people are allowed to have the same names ?
 
@Chris'ssis lol, yeah, that is really bad (deceiving). :-)
 
Also, what's the correct way to formulate that a set $X$ is not a subset of $\Bbb N$ but a subset of $\Bbb R$ thus it has the cardinality of $\Bbb R$ ? How do I prove it?
 
@Chris'ssis do @Chris'ssis
And look at the tabs
Double picture :D
 
@Huy I liked your old gravatar better.
 
4:24 PM
@robjohn This answer of mine math.stackexchange.com/questions/780913/…, which I remember you were talking about with someone on chat a while ago and seemingly didn't find any issues with, is bothering me. Does the integral on the first line actually exist as a Cauchy principal value? I'm not familiar with a definition that pertains to non integrable singularities at both endpoints.
 
@MikeMiller two requests. One, please answer my question above, two, set theory is interesting, can you suggest a solid book dealing with cardinals, ZFC, and what knowledge do I need to understand that book?
 
Huy
@MikeMiller: Not my fault.
 
@UserX what was your question above? And unfortunately I'm not all that knowledgeable about the subject. I learned what little I know from a book by Enderton, "Elements of Set Theory".
 
@Chris'ssis Don't worry I changed my name and pic
 
The reference request for set theory is for @robjohn too
 
4:30 PM
@Huy it prolly happened when you changed your email.
 
Huy
@MikeMiller: I never changed it.
 
Hmm. The gravatar is generated by stuff in your profile, so you must have changed something.
 
@MikeMiller what's the correct way to formulate that a set $X$ is not a subset of $\Bbb N$ but a subset of $\Bbb R$ thus it has the cardinality of $\Bbb R$ ? How do I prove it?
 
@UserX You can't, as mentioned above! It's entirely possible from the axioms that there are subjects of $\Bbb R$ that are in bijection with neither of those sets.
And even more practically, you use the Cantor-Schroeder-Bernstein theorem to show there's an injection from each one to the other.
 
@MikeMiller Assume I assume the continuum hypothesis
And I didn't mean robjohn
@DanielFischer can you suggest a solid book dealing with cardinals, ZFC, and what knowledge do I need to understand that book?
 
4:34 PM
@UserX Suppose you can prove $X$ is not countable, and $X \subset Y$, where $|Y| = \Bbb R$. Then $\aleph_0 < |X| \leq |Y|$, but the continuum hypothesis says $|X|=|Y|$.
 
@Nick @Parth How's it going? :D
Just so awesome here :D :)))
 
But $\Bbb \text{may be} X \subset \Bbb N \subset \Bbb R$. Does the "not countable" mean $X \not \subset \Bbb N$?
 
Any subset of $\Bbb N$ is countable; but there are countable subsets of $\Bbb R$ not contained in $\Bbb N$, too.
 
@UserX Wow, I need one too :)
 
for instance, $\Bbb Z$, or $\Bbb Q$
 
4:40 PM
@Sawarnik one what? @MikeMiller okay I get that. Why is the set $\Bbb C \setminus \Bbb R$ uncountable though? You can match $ki \to k, k \in \Bbb Z$
 
@UserX The same book :)
 
@Sawarnik yea, today I realised how interesting set theory is.
 
@JasperLoy Doesn't countable mean if there exists a bijection between N and the set. So how do I prove that there is a bijection between N and primes?
 
@UserX It contains a copy of $\Bbb R$; if it were countable, $\Bbb R$ would be countable.
@Sawarnik $f(n) = $ the nth prime
 
@UserX I knew, but I am in search of a book as well :D
@MikeMiller Well yeah, obviously.
 
4:41 PM
Then one just needs to show that there are infinitely many of them.
 
@MikeMiller oh just realised we can have fractional $i$s, nevermind I'm stupid
 
Why I didn't I think of it before? :O
 
Huy
@DanielFischer @Semiclassical: mathb.in/22879
 
Set theory is definitely interesting from whatever very little I have read.
@UserX Do tell me if you get a nice book :)
 
Here's how I learned about cardinals today; LaTeX link for chat in ucla site -> ucla main site by mistake -> let's watch that Terence Tao lecture on primes -> lol I can understand some of these stuff/read them sporadically before ->what's this $\aleph_0$ he said? ->wiki aleph null->cardinals
2
There is a name for this endless link-vortex, and a xkcd comic on it but can't remember either.
 
4:47 PM
You can see the heads of a couple of my friends in that video :)
 
@MikeMiller are those friends semi-bald with sporadically white hair?
 
The guy on the left near the start of the video?
 
Almost everyone(not to be confused with almost everywhere/almost always(that's a math joke))
 
on further review only one of those heads belongs to a friend.
 
Did you see my question at math ed.se @MikeMiller
 
4:52 PM
I just did; it's a good question
 
Thanks :D
 
@MikeMiller have you met Terence Tao irl?
 
@UserX :D :D
@IceBoy Nice :O :O
Such a nice app anyway :)
 
5:07 PM
Yes, it is cool.
 
App? :D tinder? :D
 
@TheArtist here
 
@Sawarnik hello bruce Wayne :p
 
@TheArtist hey artist :D
 
@Ice Boy ohhh :o U take pics and and the app solves it ? :o
 
5:10 PM
The artist formerly known as Prince?
 
@Sawarnik where is UR gf ? :p
 
i don't know, maybe you should ask her yourself :P :P
 
@Anastasiya-Romanova , come here, your bf is waiting for you :p
 
^Anastasiya-Romanova: Well, I have nothing to do with this! Nothing.
 
Ted? My teddy bear :p
 
5:12 PM
lelele
 
@TheArtist Delete that!
please
delete it fast :O
 
@Sawarnik , I don't see a delete option :o for real :o
 
e:18290939 Common click on the arrow which appears on the left.
jus 2 mins..
 
@Sawarnik why would she kill us? :D
 
@Sawarnik Ullo
 
5:14 PM
@Sawarnik didn't you say earlier that @Anastasiya-Romanova and you are in a long-distance relationship?
6
 
Haha :D
 
@TheArtist between the avatar and first letter of text
 
What the heck
@UserX and delete this too
 
@UserX nice one ;)
 
@Anastasiya-Romanova Please, ignore these messages! I have nothing to do with it.
@Chris'ssis Hey :D
@UserX actually no :P
[I would have liked it, but that is not the case this moment :P]
 
5:17 PM
@Sawarnik then what were those pics you shared of a asian-looking teen earlier in questionable poses and clothing claiming @Anastasiya-Romanova sent them to you?
 
@Sawarnik long distance relationships are cool...
Lol
That's too much
 
@UserX ah, can you produce some of those images? :P
:P :P
 
@Sawarnik I am a holy banana who capture the name of the unaware
 
@Chris'ssis Jasper?
 
@Sawarnik It's probably Hippa.
 
5:18 PM
@Sawarnik Check my profile
 
@Chris'ssis I know.
 
@Sawarnik it's your last deleted post, mods removed it in case it's illegal
 
@Sawarnik Check it I updated it
 
@UserX But you must have saved those images really :P
 
Huy
@TheArtist: Have you ever been in a long-distance relationship before?
 
5:19 PM
Ok, can you produce the last non-deleted message before it?
@Chris'ssis Ted, right?
@MikeMiller Oh, I see :D
 
Okay, enough fun, gotta read all the 17 tabs I have opened. Brb
 
Well done @Hippalectryon I have done the same thing with @Balarka twice :P
Ask him :P :P
 
@Huy wbu :p
 
Huy
@TheArtist: Yes.
 
@Sawarnik The delay for changing names though :c
 
5:20 PM
@UserX :D I always open 17 tabs like those, but never read it :P
 
@UserX Yes, a couple times.
 
@Huy me too
 
@Hippalectryon :P
 
@Sawarnik Whatever, next time I'll use my alt @IntegralPicnic :D
 
Huy
@TheArtist: How far?
 
5:21 PM
@Huy :O :O :O
 
@Huy in kms?
 
You didn't tell this before.
 
Huy
@TheArtist: Sure?
 
@Huy sure about? :p
 
@UserX Bye you .. grr
 
Huy
5:21 PM
@TheArtist: In whatever unit you like, as long as it's common.
 
3500 km
@Huy 3500km
 
Huy
@TheArtist: And what exactly was cool about it, for you?
 
Phew.
 
@Huy you tell me the distance between U and ur gf?
 
Huy
@TheArtist: ~1200km
Ex.
 
5:23 PM
@Sawarnik where are you from?
 
@TheArtist $\infty$ for me :D
 
@The Game hahah :D nice one bro (y) ;)
 
:-)))))
 
@Huy Ah that's close. Tell me the story ? ;)
@Huy relatively close
Where is @Sawarnik from???? :D
 
Huy
@TheArtist: I'd love to hear what exactly was "cool" about your long-distance relationship first. Not about the relationship itself, but about the fact that it was long-distance. I am currently in a short-distance relationship and cannot remember a single advantage of a long-distance relationship.
 
5:27 PM
@Huy its "is about" , not was :)
 
Huy
Sure.
So what are advantages of long-distance relationships, in your experience?
 
@Huy one advantage? :p no need to spend money on shopping , etc :p ...my purse loves it :p
wallet*
 
Huy
@TheArtist: It isn't a necessity for short-distance relationships either.
 
got disconnected :<
 
Is $\mathcal{P}(S)$ the largest discrete topology of $S$?
 
5:28 PM
@Huy Where exactly? :P
 
P(S) denotes the power set
 
Huy
@Sawarnik: That's a secret.
 
@UserX Eww, that sounds freaky.
 
@Huy my experience it is :p
 
You know all that stuff?
 
Huy
5:29 PM
@Sawarnik: Draw a circle with the given radius around the city I currently live in and find out by yourself.
@TheArtist: That sounds to me like women have only been interested in your money before.
 
@Sawarnik where are you from?
 
@Sawarnik what sounds freaky?
 
@Huy hmmmm :p ok let us hear about UR story? ;)
@Sawarnik where are you from???????
 
Huy
@TheArtist: There's not much to hear about. Just that I don't think a long-distance relationship has any advantages over a short-distance relationship.
 
And again :< :<
Let me read up.
 
5:32 PM
@Huy :) Yeah :)
 
@Huy I know :P
 
@Sawarnik where are you from??????
 
And I was just going to do that :P
 
@Sawarnik which country???????
@Sawarnik telllllllll
 
lol
ok @theartist
 
5:32 PM
@Sawarnik TELLLLLLLLLLLL
 
its a mystery for you, hint:
 
@Sawarnik I have studied some topology, now I'm reading some set theory theorems in wikipedia and I combine the theories. No I don't have any deep understanding of these, just some superficial definitions and knowledge
 
the solution is verrrryyy easy
 
@Sawarnik, what's the hint?
 
@TheArtist hint: the solution is very verrrryyy easy
may i know why are you so desperate?
 
5:34 PM
@Sawarnik there is a distance of approximately 5900km between you and @Anastasiya-Romanova
 
@TheArtist I know better :P
She does not live in Russia.
 
@Anastasiya-Romanova where r u?
@Sawarnik she's in Japan
 
@TheArtist No, I think.
 
@Sawarnik what do you mean? :p
@Sawarnik you stalk your gf?
 
@TheArtist On her true country..
 
5:35 PM
@PedroTamaroff Heya.
 
@TheArtist Probably :P
 
@Sawarnik you don't know your Gfs couhtry? :o
 
She's somewhere near Malaysia :P
 
@Sawarnik, Indonesia ? Thailand ? Singapore?
 
@TheArtist Not sure yet.
 
5:37 PM
@Sawarnik I think singapore
 
Maybe. But not enough evidence yet.
 
@Sawarnik if it's not Japan then I think it's singapore
 
She's not at all ready to disclose her location. Before Russia, it was Random country :P
@TheArtist I ll give some evidence [whatever I have] later.
 
@Sawarnik make sure you tell me when you find for sure :p
 
@TheArtist Meanwhile, where am I? :P
 
5:38 PM
@Sawarnik coz it was me who set you both up together :p
 
lol
 
it will become a joke for the whole room now
thanks to you :< :<
 
@Sawarnik , your from a country with 5 letters...I just scrolled really hard :p
 
@TheArtist :P
 
5:40 PM
@Sawarnik ;)
 
@TheArtist WHERE ARE YOU FROM?
@UserX Wow :)
 
@Sawarnik a mystery ..... you will never find
 
Nice work @UserX Teach me when you re done :P
@TheArtist Hmm, then. I can't do much about it.
But I request.
 
I remember a nursery rhyme
@Sawarnik and @Anastasiya sitting on a tree.........
 
gah :/
grr..
this artist is up to no good :/
 
5:42 PM
@Anastasiya-Romanova
 
34 mins ago, by Ice Boy
@TheArtist here
 
probably @UserX should join it to
 
Hi @Hippa :P
 
@Anastasiya-Romanova why you no tell your bf your location? :p
 
5:44 PM
@TheArtist i know why actually :P
 
@Sawarnik whyyyy? :D
 
won't tell :P
don't want a fuss on that too :/
 
@Sawarnik I have a weird question to ask lol...are you on fb or something? :p
 
Yes.
 
@TheArtist Are you?
 
5:46 PM
@DanielFischer Do you know this classic ? "every closed set in a normed vector space is the intersection of countably many open sets"
 
@RandomVariable The integrand is bounded. Why do you mention Principal Value?
 
@Ice Boy , I have been on facebook since 2006 or 2005 :p
 
@RandomVariable Oh, I see at the $\pi$ and $-\pi$ ends. I was only looking at $0$
 
@Sawarnik , read this message mathim.com/Lobby
 
No @TheArtist are you?
 
5:49 PM
@Ice Boy? Prince? O.o
 
@TheArtist i wasn't wtf anyway
 
@Ice Boy Nope....I joined MSE recently
 
@G.T.R probably
 
@Sawarnik :p
 
@TheArtist wat message?
 
5:50 PM
@Sawarnik what what?
 
fool
 
@Sawarnik I pasted the question there :p
 
nah
 
@Sawarnik did you see it?
 
@Sawarnik ?
 
5:51 PM
@TheArtist nah
 
Did I miss anything?
 
@G.T.R ok nothing.
@BalarkaSen Nah.
 
@Balarka Sen pretty much everything
 
Oh noes I missed Ted.
 
You didn't reply? @Hippa @Chris's
 
5:52 PM
@Balarka Sen , my teddy bear???
 
lol
 
@Sawarnik , how old are you ? :)
 
@TheArtist easy mystery
you fool
 
@Sawarnik who's the fool >.<
 
you
 
5:53 PM
@Sawarnik You're going to change that avatar.
 
@BalarkaSen Batman?
 
Otherwise, I am going to change my username so that you won't be able to mimick me.
 
relax @Sawarnik he is trolling you pal
 
@RandomVariable I would arrange the domain of integration so that the singularity was inside, if that is possible...
 
@Anastasiya-Romanova , your bruce Wayne is sending unwanted pictures
 
5:54 PM
@IceBoy actually no :)
 
see^
 
@IceBoy Nah, trolling is your job.
 
@robjohn did you by any chance work at Google ???
 
@BalarkaSen haha lol, that would be delicious :P
 
@Sawarnik I know it when I see it
 
5:54 PM
@TheArtist no, you fool
research before asking
 
OK, I'd just put two of you on ignore once again.
 
Both of us? :O
 
ahem two of your accounts
 
And you were going to change your name :O :D
@BalarkaSen Oh.
 
@Sawarnik , I love you....leave @Anastasia-Romanova
 
5:56 PM
2 mins ago, by Ice Boy
see^
trolling = teasing
 
@robjohn Doesn't $\frac{e^{inx}}{\sin x}$ also have a simple pole at the origin and thus a singularity in the middle of the interval as well?
 
@RandomVariable oh, wait... since $n\in\mathbb{Z}$, there are no singularities
 
@TheArtist Gah. You will make her leave the chat.
Stop this nonsense.
 
@Sawarnik put him on ignore
 
5:58 PM
@Anastasiya-Romanova he cant stand you leaving him ....so Il stop it
 
@BalarkaSen Its Hippa.
I wanted to show I am there too :P
 
@Ice Boy , what did you say -.-
 
@TheArtist For better or worse, thank you.
 
How can the rational numbers be placed to one-to-one correspondence with the integers? Did we define them that way so the continuum hypothesis holds?
 
@Sawarnik your welcome sweetheart
 
5:59 PM
@UserX No. Cantor.
 
@TheArtist I said YOU ARE A TROLL
 
@TheArtist Wat?
 

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