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

Anyone for this question ?

@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

@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?

@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.

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

@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.

@MikeMiller Yay math.stackexchange.com/a/987681/176015 he got it

So no need to worry about it

@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.

@UserX Unprovable in peano arithmetic

@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.

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.

@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?

@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

@Huy it prolly happened when you changed your email.

@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?

@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 $\Bbb Q$

@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.

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

@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

There is a name for this endless link-vortex, and a xkcd comic on it but can't remember either.

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

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 :)

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

The artist formerly known as Prince?

@Sawarnik where is UR gf ? :p

i don't know, maybe you should ask her yourself :P :P

@MikeMiller Who is this? math.stackexchange.com/users/176015/ted-shifrin

@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

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

@Sawarnik didn't you say earlier that @Anastasiya-Romanova and you are in a long-distance relationship?

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]

@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.

@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

@TheArtist: Have you ever been in a long-distance relationship before?

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

@TheArtist: Yes.

@Sawarnik The delay for changing names though :c

@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

@TheArtist: How far?

@Huy :O :O :O

@Huy in kms?

You didn't tell this before.

@TheArtist: Sure?

@Huy sure about? :p

@UserX Bye you .. grr

@TheArtist: In whatever unit you like, as long as it's common.

3500 km

@Huy 3500km

@TheArtist: And what exactly was cool about it, for you?

Phew.

@Huy you tell me the distance between U and ur gf?

@TheArtist: ~1200km

Ex.

@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

@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.

@Huy its "is about" , not was :)

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*

@TheArtist: It isn't a necessity for short-distance relationships either.

got disconnected :<

Is $\mathcal{P}(S)$ the largest discrete topology of $S$?

@Huy Where exactly? :P

P(S) denotes the power set

@Sawarnik: That's a secret.

@UserX Eww, that sounds freaky.

@Huy my experience it is :p

You know all that stuff?

@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???????

@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.

@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

@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?

@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..

@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.

@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

@Sawarnik coz it was me who set you both up together :p

lol

vimeo.com/109405701

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

@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 :/

@Anastasiya-Romanova

:p

probably @UserX should join it to

Hi @Hippa :P

@Anastasiya-Romanova why you no tell your bf your location? :p

@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?

@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?

@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?

@Sawarnik what what?

fool

@Sawarnik I pasted the question there :p

nah

@Sawarnik did you see it?

@Sawarnik ?

@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

@Balarka Sen , my teddy bear???

lol

@Sawarnik , how old are you ? :)

@TheArtist easy mystery

you fool

@Sawarnik who's the fool >.<

you

@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

@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

@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

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

LOLOLOLOL math.stackexchange.com/users/176015/ted-shifrin

@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

@UserX No. Cantor.

@TheArtist I said YOU ARE A TROLL

@TheArtist Wat?