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12:02 PM
can anyone help please? just check if my reasoning is correct
 
12:13 PM
RS agarwal
@user400188
 
@MartianCactus
Thank you.
 
12:35 PM
Goodnight everyone
 
user84215
12:51 PM
Hello everyone
 
user84215
according to ZF, How can it possible a set belong to itself ?
 
@aminliverpool it is not possible
 
user84215
I mean ZF without the axiom of Foundation
 
then how is it not possible?
 
user84215
bring an example
 
12:56 PM
just declare it to be
 
user84215
I cant understand
 
@aminliverpool just erect an axiom saying there exists a set which belongs to itself
 
user84215
I need a model in which this case happens. An example.
 
replace the axiom of foundation by an axiom declaring the existence of a set which belongs to itself
 
user84215
why dont you bring an example?
 
12:59 PM
In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by Peter Aczel (1988), as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to a unique set. In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a Quine atom. A set theory obeying this axiom is necessarily a non-well-founded set theory. == Accessible pointed graphs == An accessible pointed graph is a directed graph with...
@aminliverpool because I don't know much about alternative models of set theory
oh it's you again
you who challenged us yesterday
 
user84215
Did you reach the answer?
 
of what
 
user84215
an uncountable set without any perfect subset
 
user84215
back to previous question. I still have problem
 
are you trying to have a conversation or are you challenging us?
 
user84215
1:08 PM
I just want to solve my questions by discussing. Is this bad?
 
@aminliverpool it is not bad, but you sound like you are challenging us.
Please go on
 
@aminliverpool can you show that an uncountable closed set of the reals contains a perfect subset?
 
how the Lebesgue integration is 1?
 
user84215
this is true.
 
Hi, how can I prove that the map $f:B^n\to S^n$ given by $x\mapsto cos(\pi \Vert x\Vert)u+sin(\pi \Vert x\Vert)\frac{x}{\Vert x\Vert}$ is onto? Where $u=(0,\cdots,1)$
 
what does the lastline mean regarding f is linear on [1,1/n] and [2,2/n]?@DHMO
 
user84215
I still hve problem in viewing math formula. They appear in mathjax code.
 
@BAYMAX google "linear function."
 
yes
but in the graph its 0 ,
for [0,1/3]
 
f(x)=0 is a linear function
 
1:19 PM
@BAYMAX it's not well-termed. It just means what I plotted
 
also no f(x) does not equal 0 on [0,1/3]
 
@arctictern what?
 
f(0)=0 and f(1/n)=n so f(x)=n^2 x on [0,1/n].
 
oh
@BAYMAX sorry
I thought it said [0,1/n)
you know, the brackets are important
 
user84215
please help me. I have problem in viewing math formula.
 
1:21 PM
@aminliverpool see LaTeX in chat in room description, top right corner
 
user84215
I have done it. But the problem remains
 
@aminliverpool are you on mobile?
 
user84215
No. laptop
 
you're clicking the "start chatjax" button on your bookmarks toolbar while on this tab?
 
@aminliverpool did you use https:// ?
https can block javascripts
 
user84215
1:24 PM
I dont think so
 
@aminliverpool I know, but can you prove it? I don't think it's true for generic uncountable sets, maybe looking at where the proof for closed sets fails will help
 
user84215
First, lets speak about the fact whether that is consistent with intuition.
 
@aminliverpool intuition isn't that useful
 
user84215
But it only can develop mathematics
 
user84215
I mean math intuition
 
1:29 PM
what's up
 
@BalarkaSen welcome back
 
@BalarkaSen yo
 
Hi stranger
 
user84215
please dont forget my problem on viewing math formula.
 
@aminliverpool this is a viable alternative
 
1:32 PM
Hi @Balarka
 
@AlessandroCodenotti do you have exmaples where $\int \sum$ and $\sum \int$ give two different real numbers?
 
Hello all, please help me with my question,thanks :) math.stackexchange.com/questions/2151770/…
 
@aminliverpool Does mathjax not work on your system?
 
user84215
yes
 
user84215
@DHMO. I have done it. but no improvement
 
1:33 PM
@DHMO don't have one off the top of my head but it shouldn't be difficult to cook one up with a series that doesn't converge uniformly
 
@aminliverpool I mean, copy the mathjax into the website to render
@AlessandroCodenotti I don't see how they can give two different real numbers
I can see that maybe one converges and one diverges
 
user84215
copy mathjax? I cant understand.
 
@aminliverpool you tell me that you are viewing $e^{i\pi}+1=0$ instead of the formula
so you can copy e^{i\pi}+1=0 to the website I gave you
and to render the LaTeX there
as an alternative
 
Hm, I guess we can cheat by using an example where $\lim\int f_n(x) dx$ and $\int\lim f_n(x) dx$ are different real numbers and then defining a series so that the $n$-th partial sum is precisely $f_n$
 
@AlessandroCodenotti is the former possible?
 
1:37 PM
sure, if we're talking about pointwise convergence for the sequence $f_n$
(while you can safely swap integrals and limits if the convergence is uniform)
 
user84215
I did it and it works well on that website. but the problem remains in this chat room.
 
I have made mental peace with the fact that $\mathscr P(\mathbb N)$ is uncountable long ago. In fact, it has become so ingrained in my mind that I find hard to accept that the set of finite subsets of $\mathbb N$ is countable! — triple_sec Jun 5 '14 at 11:04
I guess I can relate to ^
 
take those functions, they converge pointwise to the $0$ function while their integral is always $1$
 
@AlessandroCodenotti ok
 
user84215
I did it and it works well on that website. but the problem remains in this chat room.
 
1:39 PM
It's easy to come up with examples to those, yes.
Think about a small triangle with height $n$ with base $[0, 1/n]$. That's the graph of $f_n$.
 
so $\int\limits_{[0,1]}\lim\limits_{n\to\infty} f_n(x)dx=0$ while $\lim\limits_{n\to\infty}\int\limits_{[0,1]} f_n(x)dx=1$
 
@aminliverpool I said it's an alternative...
 
Does anyone know how to show that half planes are non empty sets? This question has been stumping me.
 
@fluffy_muffin which half plane?
 
In which case $\int_0^1 f_n = 1/2 \cdot n \cdot 1/n = 1/2$, but $\int f = 0$ where $f$ is the ptwise limit
 
1:40 PM
Actually I thought baymax's functions are like those Balarka is suggesting, but they are different, use Balarka's functions
 
Sorry every half plane is non empty set. My book defines a half plane "loosely' imo. It defines what a convex set is then simply defines the Plane Seperation Postualte stating that the half planes H1 and H2 are the half planes associated with the line l that divdes the plane containing these.
 
user84215
How can this problem resolve in this site?
 
@fluffy_muffin can you give me an example of a half plane?
 
user84215
other suggestions ?
 
@aminliverpool you said you still have a problem with the negation of axiom of foundation
what is your problem?
 
1:43 PM
@MikeMiller I'm reading up on the gauge theory background stuff some more and I've got a small question about Spin structures. I found some statement of the form: "Let $H\to X$ be a real, oriented vector bundle and $(\mathfrak s,J)$, $(\mathfrak s',J')$ two Spin structures for $H$. Then the/a line bundle $L$ such that $\mathfrak s\cong \mathfrak s'\otimes L$(as Spin^c structures) satisfies $2c_1(L)=0$" but I have reasons to suspect that it may not be precise. Is this a correct statement?
 
@DHMO Mathematically no, I'm not sure my notation would be correct. I think an image would suffice better here. This is what my book states: s17.postimg.org/vtdpee5xb/Capture.png
 
user84215
why can we derived this fact without the axiom of foundation?
 
@aminliverpool that fact does not derive from the negation of the axiom of foundation
you need an additional axiom stating that those sets exist
 
user84215
why ?
 
removing that axiom does not immediately give you a set X={X}
 
1:45 PM
I tried using the plane seperation postualte to suppose we had two half planes. Then if the point lies i the half plane we want we are done. Otherwise, there exists a point on the line through the point in the other half plane. Then we can construct a point in the other half plane such that it is between these two points. Thus, lying in the other half plane. However, this felt like I was using what I was trying to prove...
 
user84215
without the axiom of foundation, I cannot understand why a set can belong to itself
 
@aminliverpool because there is no axiom in the remaining ZF axioms to contradict a set belonging to itself
 
user84215
show me
 
show you what?
I can't prove this.
 
user84215
ok. back to the other question. An uncountable set ...
 
1:49 PM
@AlessandroCodenotti sei qui?
 
Any ideas? Or is that actually valid proof?
 
@fluffy_muffin do you have any theorems regarding a line?
 
@DHMO The incidence axioms, segment construction theorem, betweeness etcetera. I have everything available in neutral geometry. Well up to saccheri quadrilaterals anyways.
 
@DHMO always
 
@AlessandroCodenotti oh sorry, I misread a word.
My question is cleared now
 
1:53 PM
ok
 
maybe you might want to answer the person above you
 
I already asked him or her a question related to the one he's trying to answer
 
user84215
an uncountable set without any perfect subset
back to previous question.
 
@AlessandroCodenotti heh?
 
1:56 PM
@AlessandroCodenotti no, not him
I'm referring to @fluffy_muffin
 
Ah, I don't know about axiomatizations of geometry
 
user84215
I think it is better to go out. Come back soon. Goodbye.
 
2:19 PM
Is there an explicit example of an $\alpha$ such that $C+\alpha$ contains no rationals? Where $C$ is the standard middle-thirds Cantor set
 
@AlessandroCodenotti what is the cardinality of $C$?
 
$\mathfrak{c}$, the same as the reals
 
@AlessandroCodenotti all the real numbers in [0,1] with no 1 in one of their ternary representation?
 
right
for numbers like $1/3=0.1_3$ with a finite ternary expansion ending in $1$ you use the representation ending with infitely many $2$, in this case $0.0\bar{2}_3$ and they are in the Cantor set
 
@AlessandroCodenotti assuming that $\pi$ is normal, $\pi$.
@AlessandroCodenotti I figured out that probabilistic arguments aren't good
 
2:31 PM
I'm not convinced that works
 
I have no reason to believe pi works
@Alessandro Existence is easy by measure theoretic considerations.
 
yeah, we did this problem before Balarka
 
@BalarkaSen can you instruct me?
 
you asked it to me actually. I recently learned that you can do this via Baire category too since it's a countable union of closed sets with empty interior
 
wait
isn't $C$ perfect?
 
2:34 PM
Yes.
 
@BalarkaSen do you have any answer in mind, to the rational problem?
 
What problem?
 
16 mins ago, by Alessandro Codenotti
Is there an explicit example of an $\alpha$ such that $C+\alpha$ contains no rationals? Where $C$ is the standard middle-thirds Cantor set
 
I'm also trying to find a reference for this construction since I need for my essay, but the only construction I found of a perfect set with no rationals in "counterexamples in analysis" went the direct way by modifying the construction of the Cantor set
 
@AlessandroCodenotti how would you say "you should have called me" in italiano?
 
2:38 PM
avresti dovuto chiamarmi
 
that's strange
"you have should called me"
 
chiamarmi is the contraction of chiamare me, accusative personal pronouns are absorbed by the verb if possible
 
@AlessandroCodenotti I know this
I'm talking about "aversti dovuto"
 
while the "you" is superfluous, we never have an explicit subject if the subject is a personal pronoun
 
@DHMO I can give you a Cantor set inside $\Bbb R - \Bbb Q$ but not sure of this particular sort.
 
2:40 PM
@AlessandroCodenotti I'm also not talking about that
I'm talking about "have should"
@BalarkaSen sure
 
@DHMO a lot of things are written in the opposite order in Italian when compared to English. Adjectives after nouns for example
 
@AlessandroCodenotti but adjective after nouns I've seen that in French and Spanish
 
I don't know how verb conjugations works in those languages
 
@AlessandroCodenotti in Spanish, the conditional carries a meaning of "should". Does that occur in Italian?
 
2:43 PM
@AlessandroCodenotti they're mostly similar
@AlessandroCodenotti then why did you include dovuto?
 
because it's the past form of the conditional
dovresti chiamarmi is you should call me
 
@AlessandroCodenotti can't I just say "avresti chiamatomi"?
 
no, that doesn't work
 
@AlessandroCodenotti isn't that past conditional?
 
Hello all, i need serious help here,
i want to prove that
y=sin(t^2) cannot be a solution on an interval containing t=0 of an 2nd order linear homogeneous differential equation.
using the Wronskian.
The Wronskian is 0 at t=0, but how does this contradicts Abel's Theorem?
I am stuck for a long time, please help me :(
 
2:46 PM
@AlessandroCodenotti oh, I have another question, this isn't consistent among French and Spanish: can you translate "it is I/me who said this"?
 
@DHMO Actually, I can show that there exists an $\alpha$ of that sort.
It's not particularly hard.
 
@BalarkaSen how?
 
Look at all the rational translates of $C$. That's measure 0, so is not all of R, correct?
 
@BalarkaSen oh $C$ has measure 0?
 
So pick a point from there and translate using that.
@DHMO Sure
?? irrationals have measure 1
 
2:49 PM
@BalarkaSen what?
 
@DHMO no, past conditional of dovere, for example is (io) avrei dovuto, (tu) avresti dovuto, (lui/lei) avrebbe dovuto, (noi) avremmo dovuto, (voi) avreste dovuto, (essi) avrebbero dovuto
 
you mean the irrationals in [0..1]?
 
Irrationals in R, or [0, 1], sure.
 
@AlessandroCodenotti I mean, in Spanish, "you should study" is "estudiarías"
we don't need to say "debes estudiar"
@BalarkaSen wait what
irrationals in R is also 1?
 
in Italian is dovresti studiare, we have a lot of so called composite tenses
 
2:50 PM
@AlessandroCodenotti oh, I thought you mean that dovere is not necessary
 
@DHMO lo ho detto io or sono stato io a dirlo
 
@AlessandroCodenotti couldn't you start with ___ io che ...?
 
Sorry, I meant in (0, 1)
 
@BalarkaSen I didn't know that uncountable sets can also have measure 0....
 
sono io che lo ho detto, sounds a bit unnatural though, maybe if you want to stress the I
 
2:52 PM
Sure they can
 
@AlessandroCodenotti so nobody says this? not even in songs/poetry?
@BalarkaSen any trivial examples?
 
Cantor set... :P
 
nah, it wouldn't sound weird even in spoken italian, but I'd use one of the other constructions
 
It's the easiest example
 
@AlessandroCodenotti this agrees with Spanish... in French we say "c'est moi qui l'ai dit" which would be directly "e io che lo ho detto"
@BalarkaSen measure is mind-boggling
 
2:53 PM
Nah
Not once you actually study it from somewhere other than wikipedia
6
 
@AlessandroCodenotti so in Italian there is no distinction between past tense and present perfect?
@BalarkaSen burnt
 
@DHMO there's not a perfect 1-1 correspondence between present perfect and an Italian tense I think, depends on context
there are a lot of past tenses in Italian though
 
@AlessandroCodenotti you do have the -ava tense right? Not sure how you would name it.
In Spanish we say "estudiabas"
 
that's the imperfetto I guess
 
@AlessandroCodenotti I only know two...
@AlessandroCodenotti yes, the imperfetto
 
2:58 PM
indicative has passato semplice, imperfetto, trapassato prossimo, passato remoto and trapassato remoto as past tenses (a couple of those are not widely used in modern Italian), conditional has a past tense, subjunctive has 2 past tenses
 
@AlessandroCodenotti you win
you can also say English has "I had", "I had had", "I had been having", "I would have", ...
 
there's also infinitive, gerund and participle past, but they're weird, they only have one person rather than 6
 
@AlessandroCodenotti does devo studiare mean anything?
 
I must study, rather than dovrei studiare, which is I should study
 
this is really strange
I thought they mean the same
$\text{I must study} \iff \text{I should study}$
 
3:01 PM
as you said conditional has a "should" connotation
 
If I am not wrong, in Spanish debo estudiar = estudiaría = I should study
 
I'm don't know about Spanish. But they are different in Italian, at least in some contextes, for a practical example say that you ask me whether I want to come to the cinema tonight. "I can't, I must study" would be non posso, devo studiare while "I should study, but I'll come" would be dovrei studiare, ma verrò
 
@AlessandroCodenotti interesting
@AlessandroCodenotti I guess I would say no puedo, necesito estudiar
 
oh, nice, pointwise limit of continuous functions can be bad but not that bad, it must be continuous on a dense set at least
 
and estudiaría, pero vendré
@AlessandroCodenotti a limit is continuous?
 
3:08 PM
a limit of a sequence of functions is a function
 
@AlessandroCodenotti Yeah
 
@AlessandroCodenotti ah, nice
@AlessandroCodenotti how would you say acabo de comer = I just ate?
 
I wonder why Munkres calls this fact "more amusing than profound"
 
(In English it's in the past tense but everything is in the present tense in English)
 
@DHMO Ho appena mangiato
 
3:10 PM
@AlessandroCodenotti no present tense formation?
I just checked that Spanish has apenas comí also
 
Is this an application of Baire category
 
nope, it's a past action after all. The "appena" signals that it just ended
 
@AlessandroCodenotti you can see this
I guess it's idiomatic
 
@BalarkaSen yeah, it's not an immediate one though, I have yet to read the proof in detail
 
but in French we also have je viens de manger = vengo di mangiare
 
3:12 PM
doesn't work in Italian
 
interesting
 
@AlessandroCodenotti Well, pointwise convergence is like, dumb and nobody cares about them
 
@AlessandroCodenotti could you read comer = mangiare without the translation?
 
It's not the right topology in the space of functions at all
 
3:13 PM
but it's the product topology on $\Bbb R^\Bbb R$ I think?
 
Yes. And product topology is super-dumb in that context
 
@DHMO nope, comer doesn't look like anything in Italian
 
But...why...?
What has Jennifer lawrence had to do with group theory?
 
@BalarkaSen Fair enough
 
@AlessandroCodenotti which gives rise to Cómo como? Como como como. = Come mangio? Mangio come mangio
 
3:15 PM
Dont we get MSE T shirts ??
 
@Secret welcome back
 
that's a funny sentence
 
@AlessandroCodenotti Italian ranks at the 18th in terms of population of speakers...
whereas Spanish ranks at the 3rd
@AlessandroCodenotti how do you say "July 4"?
 
we write 4 Luglio and read Il 4 di Luglio
 
@AlessandroCodenotti expand 4 please?
 
3:18 PM
Product topology is never the right thing for infinite products; the open sets are huge. Indeed, I think box topology (the correct alternating for product topology in some cases) is finer than uniform topology.
 
quattro, non quarto (four rather than fourth)
 
Hey everybody relax as i am BAYMAX!!!
 
@AlessandroCodenotti I see
do you trill the r in both words?
 
yep
misread what you wrote
 
3:19 PM
@AlessandroCodenotti cuatro, no es cuarto
 
wait what do you mean with the uniform topology? I need a space of functions with codomain $\Bbb R$ or can it be generalized?
well to linearly ordered sets probably
 
I came across this fascinating and humbling sentence:
> The set $V_6$ contains $2^{65536}$ elements, which very substantially exceeds the number of atoms in the known universe. So the finite stages of the cumulative hierarchy cannot be written down explicitly after stage $5$.
(regarding the von Neumann universe)
 
@DHMO consider a $10\times10\times10$ lattice. On each lattice you have a spin that can either be oriented $+1$ or $-1$. In statistical physics the expressions to get an expectation value involve summing over all possible states, for such an extremely small and easy lattice, how many terms must you calculate to find the expectation value?
What I wrote is actually wrong, there are many tricks to calculating the expectation values of that you can actually get analytic expressions easily in this case
but if you want to find the ground state of a quantum spin system analytically there is almost no way of going around this gigantic number, even for this super tiny lattice that is way to small for solid state applications ($10\times10\times10$? you really want something like $10^6\times10^6\times 10^6$ size lattices..)
 
@AlessandroCodenotti how on earth do you distinguish the two pronunciations of e?
 
there are at least $3$, è, é, and e, but a lot of people pay no attention to the first $2$. The difference between an e and an accented e is very strong to me though
 
3:32 PM
@AlessandroCodenotti for example?
bene?
 
that's hard to explain by writing, you should look at some recordings of Italian speakers
 
@AlessandroCodenotti IPA?
 
I don't know how it works well enough to write it
 
alright
could you give me a word though
and I can look for recordings
Do you distinguish beve vs bene?
@AkivaWeinberger hola
bienvenido
 
no, it's mostly at the end of words. You should look at words ending in e and words ending in è or é
 
3:36 PM
Hola
 
@AlessandroCodenotti can you give me an example?
@AkivaWeinberger we're discussing Spanish vs Italian
 
also look at the difference between e, which is the conjunction and, and è which is a form of the verb essere
 
(mostly it's me annoying him with questions about Italian)
 
that works for other vowels too, pero is a pear tree, però means but, papa is pope, papà is father
 
@AlessandroCodenotti I hear "questo è" being pronounced as quest'è
@AlessandroCodenotti però vs ma?
In Spanish the former is used much often
 
3:41 PM
sometimes we do skip the last vowel of a word if the following one begins with a vowel too, quest'anno sounds better than questo anno, I wouldn't skip it in questo è though
I think they have the same meaning in most contextes
 
In Spanish mas seems very old
 
This "skipping the last vowel" was very common in Latin poetry :O
 
@SteamyRoot hmm...
for example?
 
Well, take my favourite poem "Odi et Amo" by Catullus
 
in where I live... it's the 7th case of secondary school student suicide this month....
 
3:44 PM
You'll note that every vowel has something above it, either a curve line (meaning "short") or flat line (meaning "long")
 
none of you probably cares about where I live though
 
the ones that don't have anything, are actually absorbed by the vowel next to them
So you'd say "Od'et amo, quar'id ... senti'et ..."
 
@SteamyRoot What I don't like is that the spondee and the [insert name] don't follow the word boundaries...
it makes keeping track of the rhythm very difficult
 
Well, the rhythm is very different from "modern" poetry :P
 
compare it to iambic pentameter
 
3:47 PM
They're dactylic penta- & hexameters :P
 
Did anybody else just get pinged about the even in hbar?
 
Not that I noticed
 
@DHMO Oh my god, that's horrible
 
@AkivaWeinberger it is...
considering the smallness of our city...
 
Hong Kong, right?
 
3:51 PM
even one case is too many..
@AkivaWeinberger yes
while all the Westerners know is that asians are somehow gods
 
I've heard about student suicides in Korea as well
 
I've heard that the students in Korea have much more pressure than in Hong Kong
 
The suicides seemed to be close to the start of the TSA or something similar in March. Just like what happened last year where there are 16 something cases of suicides
 
@Secret oh, I forgot that you are also from Hong Kong...
but that was 16 cases in a year
this time it's 7 cases in a month
 
Those 16 cases, if you read closely, really only started to escalate during the start of march. So it is possible they are correlated to the TSA and the pressure stemmed from it
 

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