« first day (2185 days earlier)      last day (1731 days later) » 

3:02 PM
Ok I have just read it, this is really strange. So basically, the volume does not change but things getting further apart because the metric changes...?
> In differential geometry, the backbone mathematics for general relativity, a metric tensor can be defined which precisely characterizes the space being described by explaining the way distances should be measured in every possible direction. General relativity necessarily invokes a metric in four dimensions (one of time, three of space) because, in general, different reference frames will experience different intervals of time and space depending on the inertial frame. This means that the metric tensor in general relativity relates precisely how two events in spacetime are separated. A me
Now that I found the term, I should see if there are any PSE questions on it
Q: Metric Expansion Of Space

rahulgarg12342I just do not understand this concept of metric expansion of space. Shouldn't the galaxies move away from each other. How can the space between them expand if the galaxies are not moving away from each other. I do not want an analogy of a balloon or something like that. I just want a real answer ...

@Slereah, if you are still somewhere around in space, THIS is what I am trying to ask whether metric expansion implies there is more space, and if it isn't how to physically interpret metric expansion\
@Secret: spacetime is not an object. It doesn't stretch and new spacetime isn't created.
When you have a time dependent metric, like an FLRW universe, it means the proper distance between two points may change with time. But this doesn't mean anything is stretching.
You're looking for a physical mechanism that simply isn't there.
So you mean the accurate picture is that things getting further away from each other without anything stretching at all, and that is the correct way to understand a time dependent metric?
@Secret yes. The proper distance between points in space is not necessarily a conserved quantity in GR.
3:19 PM
Hmm I see... thanks
In that case, looking back to that tardis question, then (at least mathematically) to create a tardis space can be as straightforward as simply increase the spatial components of the metric in some finite region of spacetime, and the bag of gold spacetime will be one where the metric is FLRW thus it increases forever in a confined region (in the naive sense)...
I do find the voting on your and John's answers quite interesting, btw - it seems this question may well hit the precise sweet spot where 50% of the site users think it's OK and 50% think it goes too far (or at least without proper attribution). — Emilio Pisanty 9 mins ago
@Secret that's a specific geometry called a Friedmon, made by joining together two metrics. There is no physical motivation for doing this and no plausible mechanism for how such a spacetime might be formed. Because you can do it mathematically doesn't mean it can exist in practice.
Vote for me! I will give free doughnuts to everyone voting for me¹.

¹ Recipient pays shipping. Restrictions apply.
3:28 PM
In other news, I have finally crunch through this lengthy article (though I still understood nothing about homotopies)
In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, and an orthogonal matrix is a real matrix whose inverse equals its transpose. The determinant of an orthogonal matrix being either 1 or −1, an important subgroup of O(n) is the special orthogonal group, denoted SO(n), of the orthogonal...
@JohnRennie I did upvote; however, what DavidZ has said is not wrong too.
25 mins ago, by John Rennie
@Secret: spacetime is not an object. It doesn't stretch and new spacetime isn't created.
This is actually the 2nd time I made this mistake since the days of Volderfield, hopefully I will not foget the explanation in the future and stop mking this mistake. I think my mind have not adapted to the weirdness of spacetime yet, despite seemed to be ok with the weirdness of quantum
The point is you need to cite when you are using some external resource.
@MAFIA36790 are we writing academic papers here or answering questions? For an academic paper, I would heartily agree that this qualifies as plagiarism and should be handled
on PSE I would only be offended if someone copy&pasted other websites
perhaps, after reading this I should have a much better intuition for GR related stuffs...
3:37 PM
I think it applies to both.
@MAFIA36790 If it had been me writing the answer I would have cited the article, and in any case I would have found my own words and not used the words from the article so freely. However I don't want to put CountTo10 on trial.
[which does not mean that it would be wrong to name the source]
7 hours ago, by MAFIA36790
> Do not copy the complete text of external sources; instead, use their words and ideas to support your own.
He didn't copy the complete text.
3:38 PM
But he didn't cite.
That's why an accusation of plagiarism is going a bit far. But I agree that if it had been me I would have cited the article.
I say ban the guy.
It has been explicitly mentioned in the plagiarism help page that you need to cite as well as not copy the exact words.
@JohnRennie the question is not "would we have cited" but "is it necessary to cite in order to prevent deletion of the post", isn't it?
Actually meh is a pretty good response. This whole thing is a real storm in a teacup.
3:40 PM
well, it is good that it surfaced this time; it will set a remainder to cite when your post is based on past works.
I liked though how JohnR differentiated in the first para the plagiarism in academia and that here.
Well, actually, I've seen the user CountTo10 around, and they are actually incredibly active, going through the queues and everything. Banning would definitely not be an appropriate response. And I'd have to agree with Sir Rennie in this case. (@JohnRennie, I'm not going to pay for the shipping to have a donut, but I did upvote your answer. =)
@Sanya Well, I don't think we need to start a manhunt on the 'plagiarist' in this case, but I would prefer that if you are basically rephrasing another source, you mention that.
@heather Banning?
@MAFIA36790, 0celo7 said to just "ban the guy".
3:44 PM
No one is banning anyone ;P
Plagiarists are only suspended in egregious and repeated cases.
@heather sadly the restrictions prevent the shipping of doughnuts to minors on health grounds. It's for your own good :-)
@heather He sometimes ..... Well, let it go.
@heather Maybe...don't take everything he says at face value ;)
But I will eat your doughnuts for you ;-)
3:44 PM
@MAFIA36790, I know they wouldn't be banned, it was just more of a defense of the user.
hey folks, did anyone see this?
> Wigner Functions for Arbitrary Quantum Systems

> The possibility of constructing a complete, continuous Wigner function for any quantum system has been a subject of investigation for over 50 years. A key system that has served to illustrate the difficulties of this problem has been an ensemble of spins. Here we present a general and consistent framework for constructing Wigner functions exploiting the underlying symmetries in the physical system at hand. The Wigner function can be used to fully describe any quantum system of arbitrary dimension or ensemble size.
@ACuriousMind, if I took everything 0celo7 said at face value I think I'd be pondering whether or not I exist right now =P
and get any real understanding of what's going on?
@JohnRennie, thank you for the eating of the donuts for me ;)
@heather Any time. I can eat another if you want ... :-)
3:46 PM
ricekrispies are better than donuts.
@JohnRennie, sounds good to me =) make sure it has chocolate frosting and sprinkles, those are the best
@JohnRennie sure ;)
@obe What!!!
so, I think I got my first review "audit"
(I was going through documentation changes on stack overflow)
I did pass, if you're wondering
A: How can KVL & KCL be derived from Maxwell equations?

Sambeet Panigrahi@Tobias.Thanks for clearing this.I came across something that I would like to add to this discussion.Voltage is actually a very well defined concept in electromagnetics.It is the retarded electric scalar potential in the Lorenz Gauge as defined in this report. This report is also consistent with...

3:49 PM
Not an answer?
@JohnRennie depends on the donut?
@EmilioPisanty, 3 volumes, wow. What are the prerequisites?
@heather what, for Spivak?
@EmilioPisanty, yep
It's actually five volumes if I recall correctly
3:50 PM
@heather Hmm? We don't have review audits on physics.SE.
@EmilioPisanty, oh, really? Geesh.
but you can restrict yourself to the first volume and have a perfectly happy time
@ACuriousMind, I was doing documentation reviews on stack overflow.
@EmilioPisanty, ah, okay. Does each volume cover something different, then, or...?
3:51 PM
Volume 2 is a lot heavier going - it's an in-depth reading of the original papers by Gauss and Riemann
@ACuriousMind what I'm asking is - do we prefer or do we require to
I don't know what's in 3,4,5
but volume 1 is plenty good
You do need some pretty strong calculus, though
@ACuriousMind, okay, quick question, it shows that there are 3 things in the review queue but once I click in it says there are no things to review...(stack overflow queue)
@EmilioPisanty, hmm, I might have to hold off on it then. I'm teaching myself calculus now. How well would both volumes of Apostol prepare me for it?
@heather after both volumes of Apostol you'll probably be relatively good to go as far as formal prerequisites go
it does assume a certain level of mathematical maturity
@Sanya I think that's what John's and David's answers are basically there to decide - John's saying it's fine not to leave a reference and David is saying it should be there. I'm inclined to agree with David but I am mindful that it might be difficult to determine where the line between paraphrasing something and writing in your own words is when it comes to technical writing.
3:54 PM
and that's more of an intangible
@heather SO queues move very fast, it's entirely possible that the three items just were reviewed in the time your computer took to load the review
@ACuriousMind yup. Also, there's times when there's only so many ways to correctly phrase something.
@ACuriousMind, yeah, but I clicked back out and it still showed 3 in the queue. Click back in, nothing. Click back out, there.
This isn't quite there, but it's not that far either
@heather Maybe you're not qualified to review the items that are in there
I.e. you need more rep. I don't know what threshholds there are on SO
3:55 PM
@ACuriousMind I would then start a separate discussion because as of now, Davids post just states why it should be considered plagiarism and Johns says why it shouldn't be punished ...
@ACuriousMind I'm not sure I'm saying it's fine not to leave a reference, I just saying that it isn't grounds for deleting the question and burning the OP at the stake.
@ACuriousMind, oh, huh. I know there was some idea floating around about basing reviews off of tags, but I don't know if it was ever carried out.
@EmilioPisanty, then I think I'll hold off on Spivak. I really don't think I have much mathematical maturity at all, and if I've just learned calculus I'll be struggling through it, I think, from the way you described it.
Hey Hey @MetaEd.
That's five mods in the room!
@heather My recommendation is to get it after you're done with Apostol, if you're at all interested in differential geometry. Have a look through it at that stage. What might happen is that you'll find it mystifying - as in "why are they doing this to begin with?", in which case you can just shelve it and return to that book later.
3:58 PM
@MAFIA36790 Mornin.
@Sanya I'm not sure I follow: David is saying it's plagiarism (and by inference should be dealt with like other plagiarism, either editing in the attribution or deleting the answer) and John is saying it isn't plagiarism and we shouldn't do much, if anything, about it.
@ACuriousMind no, I do not read Davids post the same way as you do
@EmilioPisanty, ah, okay. Seems like a good idea to me. Thanks for the recommendation and advice!
this inference does not seem implied in his post to me
@Sanya Read the comments, where Emilio asks basically about that.
4:00 PM
@Sanya I read David's answer much like ACM - he's saying that the similarities are enough to qualify for plagiarism, at least as I read it.
well, then I hope I'm the only one misreading the post
Well, he was not using the exact words; but the blow came when he didn't cite the link. DavidZ wrote how to modify the post like using paraphrasing the source... which is also evidently clear in the plagiarism help page.
@Sanya well, how do you read that answer then?
Plagiarism isn't a binary property. It's a scale from gratuitous to not at all. So I'm not sure it's helpful for David to come out and say (as he does) this is plagiarism.
@EmilioPisanty as a statement that and why it constitutes plagiarism without any indication of what the correct way to handle this would have been
4:06 PM
okay, one last resource question: what is a good introductory book on discrete mathematics?
If ten is gratuitous plagiarism and zero is not at all then I'd probably score the post in question as a five. The question is to have a similar sliding scale for our response, and if so where on that scale does this question lie?
@JohnRennie, gratuitous plagiarism? what is that?
@heather roughly: an answer that is entirely a copy-paste from wikipedia, with no link or attribution
@heather Gratuitous plagiarism would be verbatim copying of someone else's work with no attribution.
@JohnRennie, I see, and 0 would be not at all plagiarism on this scale
4:09 PM
So then I guess I'd rate that answer closer to 0, maybe more like a 3.5
(though I guess that's just me)
@JohnRennie We don't really have much possibility to have a sliding scale of response, imo. I see only four possible actions: No action, editing in attribution, deleting the answer, deleting the answer(s) and suspending the user. Since I would reserve suspensions for things like all of your answers are plagiarized or you have been asked to provide attribution multiple times and still don't do it, there are three actions: No action, editing in attribution, deleting the answer.
@ACuriousMind there's a sliding scale in terms of how you notify the user in question, and how long you give them to respond before (the attribution is edited in)/(the answer is deleted)
@EmilioPisanty exactly that - how about leaving a comment and waiting a few days?
@Sanya Even through if the user is active?
4:15 PM
@Sanya "a few days" seems excessive to me. I would set the maximum waiting time at floor(24 hours, 8 hours after the user sees the notification).
Waiting for responses from individuals is not how SE typically works. For instance, there's explicit guidance on mother meta that in general, all actions on a post should refer to its current state, not what it might become
Since almost all actions can be reversed, there is little harm in taking immediate action.
@ACuriousMind I disagree
Immediate deletion can and does cause harm
@ACuriousMind besides being very unfriendly :/
case in point the answer at hand
deletion of the account.
4:17 PM
@MAFIA36790 nah, but sometimes they might not be
Particularly because it takes a very experienced user to know that they can challenge a deletion.
Instantly; although that was a bit over-reaction.
@EmilioPisanty In this case the harm arises from the action taken being perceived as the wrong action to take at all, not from the action being immediate
Q: What would the universe look like if it was not flat?

SuaveSourisTitle says it all. I'm just curious to hear any wacky consequences. Assume I know little on the topic--because that's true.

And someone has upvoted it!
@JohnRennie What if?
4:21 PM
On the other hand, this is a calculation i've been meaning to do for ages ...
Q: What distance apart should two objects be placed such that gravitational forces are cancelled by an expanding universe?

BurnsBAAssume an empty universe, but one that is expanding the same way ours is at "74.3 kilometers per second per megaparsec" 2 or $$ 7.43 × 10^4 \ m\ per\ s\ per\ 3.086 × 10^{22} m $$ Which I think is $$ \frac{ 2.41 × 10^{-18} m }{ s · m }$$ Call this $ R_p $ (rate of expansion), but assume this...

@ACuriousMind Yes. But a policy that doesn't take into account that mistakes happen (and thus fails to provide suitable buffers for them) will inevitably run into that again.
@JohnRennie making the units upright...
I don't think it's hard as there's a simple expression for $\ddot[a}{a}$, but it needs some care to define exactly what you are asking.
@MAFIA36790 done
4:24 PM
@EmilioPisanty But waiting for a response effectively means that users could with impunity err on the side of not including attribution because if needed (i.e. if someone finds their source), they can always do that later
As I said, I think that the fact that almost all actions can be reversed is already sufficient to deal with mistakes
@ACuriousMind I don't see why a few-hours waiting period changes that. You're still dependent on someone catching the source, and the "they can always do that later" also holds for cases where (action is taken → post is fixed → action is reversed)
@ACuriousMind For editing, yes. For deletion, it's not clear at all to a new user that it can be undone, or how they would go about that.
↑ grrrrrrrr
@EmilioPisanty Okay, those are fair points. I'm still a bit uncomfortable with the "wating for a response" because it's an action that targets the user, not the post.
why leave the n out of the $ if you're gonna $ the \omega?
The shorter minus at the third one is even more irritating, I think
@ACuriousMind I don't see it that way at all. It's a policy that recognizes that action needs to be taken but that it's beneficial for everyone involved if it's the OP themselves that takes said action.
@ACuriousMind yeah, that too.
4:32 PM
Hello fellow humans
and ACM
Do you understand analysis now?
@0celo7 No, I need to read abbot
@ACuriousMind Hey, what's a non-Borel set in Rn
@EmilioPisanty I'm not sure I see the benefit. Do you mean the benefit that in the case of a mistake we don't get a meta discussion?
@BernardMeurer Greetings, human.
@BernardMeurer how'd you do on your QM exam?
4:35 PM
@0celo7 It is consistent with ZF that there are no non-Borel sets.
@0celo7 It was well I think, grades aren't out yet. Yours?
You need the axiom of choice for one to exist.
@ACuriousMind Yes. Also, it does not present an unduly harsh face to new users.
@ACuriousMind Right, so I heard. Proof idea?
Remember - assume good faith. I.e. assume that it's a mistake on the part of the OP.
(at least in the absence of previous occurrences)
4:36 PM
@ACuriousMind let's assume I accept the devil's axiom
What would such a set look like?
@0celo7 Look here
Discussion of plagiarism at work here I see. I just drafted a plagiarism meta post for EL&U and wouldn't mind if anybody wanted to read it and tear into me.
What the fuck you have have R = countable union of countable sets
@ACuriousMind how is such a thing possible?
4:40 PM
@0celo7 That's one of the fucked up things you get when you don't accept choice, I'm afraid :P
@MetaEd Link?
@EmilioPisanty Yes, that makes sense.
@MetaEd We needed you the other day and you didn't answer the call dawg
@ACuriousMind $\mathbb R$ isn't necessarily uncountable in ZF\C?
man, math is messed up
@BernardMeurer, further victory - I created the NOT gate function. =) I'll update the github code.
4:42 PM
@heather NOT? Ya mean CNOT?
Q: What is plagiarism? How do I avoid it? How do I address it when I see it?

MetaEdThe purpose of this question is to clarify how plagiarism is defined for the English Language & Usage Stack Exchange, and how the community can respond appropriately to plagiarism when it is discovered. This is being posted for discussion, but my hope is that after getting feedback, correcting e...

@BernardMeurer, nope, NOT. I'm doing single qubit gates first.
@MAFIA36790 That's the one.
@heather Ah, smart path :)
@EmilioPisanty Yes (I don't understand the proof, though ;) )
4:43 PM
@BernardMeurer Guess I missed the memo.
Or, well, $\mathbb{R}$ is uncountable
@ACuriousMind that's so bizarre
But without choice you can't show that the countable union of countable sets is countable
@MetaEd Fort Worth? Cool spot
@BernardMeurer, yeah, I'm taking care of the easier stuff first.
4:44 PM
Because the construction of the counting function uses choice.
@ACuriousMind ah
So it's possible that that union is uncountable, and one can apparently construct models where this is indeed the case for the reals
@heather Good, you're smarter than I am when it comes to projects :p
@BernardMeurer Not in the summer!
Yeah, I was going to say, I'm really struggling to see how Cantor's diagonal argument fails without the axiom of choice
4:45 PM
@MetaEd I come from Rio de Janeiro, Brazil. You live in Alaska for me
@ACuriousMind but this sounds plenty reasonable on the other hand
We get 43C in the bloody Winter
@ACuriousMind Yeah.
@BernardMeurer, not so much that, it's just I'm still not sure how to set up the two-qubit stuff; I'm thinking about it.
So...are you telling me that every time I take a countable union of countable sets, and get a countable set, I'm using Choice.
4:46 PM
@0celo7 yes
have a go through the argument
@BernardMeurer You'd like Texas in the summer then.
I know the argument
But is it equivalent?
@0celo7 "equivalent" is a strong word
No, I think this is strictly weaker than full AoC and probably what's called "countable choice".
Ah, ok
That's fine
4:48 PM
Better ask your local logician for conformation, though
@MetaEd Lisbon has the perfect weather imo, it's never warm, it's never cold
Or actually, I'm not so sure anymore
@BernardMeurer Plus Portuguese.
@MetaEd What about it?
4:48 PM
Portugese is just Spanish
@0celo7 Think about Bob in skinny jeans
You should go to Norway @bernard; there is rain even in winter.
@0celo7 Does that image not traumatise you?
"a countable union of countable sets" is a set $S$ whose elements are sets, endowed with (i) a bijection $f:\mathbb N \to S$, and (ii) for every $R\in S$, a bijection $g_R:\mathbb N \to R$
↑ right?
4:50 PM
@MAFIA36790 Norway is too cold
No, why would it?
@BernardMeurer I love such weather.
Not this tropical humid sub-continent weather ;/
@EmilioPisanty Not quite, the union of sets is not a set of sets, it's a set with all the elements of the original sets
@BernardMeurer It's a pretty language.
4:52 PM
$A$ is not an element of $A\cup B$
@0celo7 so what is a countable union of countable sets?
@MAFIA36790 What about rain in winter is remarkable?
@MAFIA36790 Yeah, but your weather is shittier than the on I had in Brazil though :p
I feel for you?
I'm not saying the union is the set of sets, I'm saying it presupposes the existence of such a set
@MetaEd You think so? I never liked my language (nor have I met someone who does)
4:53 PM
@EmilioPisanty You need a notion of infinite union first: given a family $\{U_\lambda\}_{\lambda\in\Lambda}$ we define $$\bigcup_{\lambda\in\Lambda}U_\lambda=\{x\mid \exists \lambda\in\Lambda:x\in U_\lambda \}$$
@EmilioPisanty That's what you need to define such a countable union, yes. The union itself is then the union of all elements of $S$.
Then take $\Lambda=\Bbb N$
@0celo7 Yes. But then you can define $S = \{U_\lambda | \lambda \in \Lambda\}$
Oh, that's what you're doing. Then I guess they're the same?
@BernardMeurer How many musicians do you know?
4:55 PM
but that's not really a union
@0celo7 They're not the same
but the set of sets exists given the union
and that's all you need
you wrote down my $\{U_\lambda\}_{\lambda\in\Bbb N}$
@MetaEd That don't speak Portuguese? None. Or do you mean how many do I listen to?
@EmilioPisanty Yeah, a countable union is a union of countably many things
So we're on the same page
@BernardMeurer Outside Brazil or Portugal, people who encounter the Portuguese language are mostly musicians encountering Portuguese song.
Particularly Brazilian song.
4:58 PM
OK, cool. So take my $S$ above. Then set $F(1) = g_{f(1)}(1)$, $F(2) = g_{f(1)}(2)$, $F(3) = g_{f(2)}(1)$, $F(4) = g_{f(1)}(3)$, $F(5) = g_{f(2)}(2)$, $F(6) = g_{f(3)}(1)$, $F(7) = g_{f(1)}(4)$, $F(8) = g_{f(2)}(3)$, $F(9) = g_{f(3)}(2)$, $F(10) = g_{f(4)}(1)$, $F(11) = g_{f(1)}(5)$, $\ldots$
@MetaEd I really like Brazilian music, alas I'm biased
In particular Bossa Nova and Jazz
That's not going to be injective
That is, you're guaranteed orderings in $S$ and inside each element of $S$, so you have your grid, and then you do your growing antidiagonals
assuming I'm reading this correctly
@0celo7 doesn't matter, you get a surjection $F:\mathbb N \to U = \cup_{R\in S} R$
5:00 PM
@ACuriousMind No, cold weather.
oh you're defining maps backwards
We have Westerlies in North-East India; means rain in winter. Not a bad experience though.
I was thinking $F:U\to\Bbb N$
@BernardMeurer you should.
5:02 PM
@Secret I'll happily buy a beer for whoever came up with that name
@0celo7 no, I don't think you need that
@EmilioPisanty Ah, holdup. Here's the kicker: When we say a set is countable, we only mean that there exists a bijection to the naturals, not that that bijection is fixed as $g_R$. Given a countable set of countable sets, choosing one $g_R$ for each $R$ is precisely what you need the axiom of choice for.
@EmilioPisanty you don't
@MAFIA36790 That ? was a typo :p, I do feel for you
It said there in the link: drop the s in spin groups to get pin groups. spin groups are a subset of pin groups
@ACuriousMind Damn, you should be a lawyer
5:03 PM
if there is a surjection $F:\mathbb N \to A$ then $A$ cannot be uncountable.
the name arise from the pun of SO to O as spin to pin
Really, all logicians could be lawyers and have much better lives
@EmilioPisanty Yeah, yeah.
@ACuriousMind ah. That's what it is.
math is messed up
I guess you could change the definition of a countable set to carry a fixed bijection with it, but then you probably run into even more horrible problems
@ACuriousMind yeah, that's gonna do pretty weird things to math
5:04 PM
Right now I am reading spin groups. The home of spin and spinors itself. Hopefully I don't need too much clifford algebra to understand it: en.wikipedia.org/wiki/Spin_group
If nothing else, it probably means that you won't be able to build that model in math.stackexchange.com/a/140582/30935
@ACuriousMind Right. Then any time you construct some other set you'd have to modify the fixed counting function
i.e. you won't be able to show that each of the individual sets of the union is countable, because you won't be able to assign it a canonical ordering
That doesn't work. The finite Union of countable sets would not be countable
Because the new counting function would be some artificial one, not the god-given one
man, every time y'all discuss the axiom of choice I take a hit on the part of me that thinks math makes sense
I'm also pretty sure I wasn't warned this required Choice when I was taught it in undergrad
5:09 PM
@EmilioPisanty Yeah, "making sense" becomes a pretty hopeless thing to hope for once one descends into the madness that is the foundation of set theory.
@ACuriousMind =(
so, hang on, remind me again why the AC is bad?
@ACuriousMind Ok, status: I reject AoC but accept C-AoC
ZF is fine.
So what is ZF + C-AoC?
@EmilioPisanty I'm of the opinion that it's not, but other people find things like the Banach-Tarski paradoxon or the well-ordering theorem...unacceptable
@0celo7 what's C-AoC?
Countable choice.
Not sure if that's the "official" notation for it.
Countable choice seems reasonable.
5:12 PM
I think one common notation is ACC, so it would be ZF + ACC or ZFCC, probably
If I can count something, then I should be able to choose things while counting.
But a choice function in the uncountable case...I don't see why that should be possible.
@ACuriousMind You're forgetting Tychonoff too (@EmilioPisanty)
Wth is a "paradoxon"
@0celo7 which one's that?
@EmilioPisanty Arbitrary product of compact spaces is compact
@0celo7 meh
I'm not invested enough in set topology to find that upsetting
@0celo7 That's me not using the correct English word :P
5:16 PM
The one that gets me is that if $V$ is an infinite-dimensional vector space, then you need AC to show that its algebraic dual is nonempty.
> The restriction to the even subspace is key to the formation of two-component (Weyl) spinors, constructed below.
Going abstract does help understand the physics
I'm 99% sure that the German word is Paradox.
@EmilioPisanty Algebraic or topological?
Or maybe both
i.e. "For every infinite-dimensional $V$, $V^*\neq \emptyset$" is equivalent to AoC
@0celo7 algebraic, as I recall
Q: Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?

Konrad SwanepoelIf $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The existence of a non-zero linear functional can be shown by taking a basis of $V$ and specifying th...

↑ that's the one
@0celo7 Paradox and Paradoxon are interchangable in German.
I've seen all these AC posts on MO a billion times
But I'll be dammed if I know what they're saying
@ACuriousMind tl;dr on forcing?
5:21 PM
It's a magic technique to prove results in logic/set theory :D
> if {\displaystyle i\neq j} i\ne j. This anti-commutation turns out to be of tremendous importance in physics, as it captures the spirit of the Pauli exclusion principle for fermions.
@ACuriousMind Is it PhD level?
So pauli exclusion principle can be captured by the anticommutation due to the clifford algebra, wow
@0celo7 I have no idea, since I don't know what would be considered undergrad logic/set theory since there are no real logicians here.
But given that it's only been discovered in the 60s, I guess it's rather advanced
HE was discovered in the 60s, and it requires basic algebra and calculus.
i.e. high school level
5:30 PM
best intro to gr.
Mathematics anyone?
Q: Does infinity have any physical significance?

Aaron AbrahamFirstly, I believe it would be prudent to mention that I've already seen this question, and no it doesn't help my case. Has infinity been observed yet? Anyways, As of late (stemming from a recent series of Physics laboratory experiments we had at school over the week), I've slowly become disi...

@AaronAbraham You need to be much more careful with terminology when you want to discuss the formal meaning of "infinity" because there are many different concepts that all can be called "infinity" and that all have different meanings. For example, there is a mathematically well-defined meaning to writing something like $\lim_{n\to 0} \frac{1}{n} = \infty$, which would colloquially be "1 divided by 0 is infinity", and this has nothing to do with physics, as you claim.
Many (though not all) of the occurences of infinity in physics can, with varying degrees of effort, be rendered mathematically well-defined and meaningful, so your question is not really answerable in the vagueness in which it is posed
But, certainly, it is not the case that the use of infinity is a "sham" by physicists in those cases
It's just that physicists rarely bother to provide the full mathematical framework, which you can take issue with, but which isn't really particular to infinity as such, since that also happens with many other mathematical concepts
Ah well...
@ACuriousMind Could you make that an answer? I'd like to think about it a bit more. Thanks!
5:49 PM
@BernardMeurer I'm wearing your favorite shoes
@0celo7 Oh god, no
which shoes?

« first day (2185 days earlier)      last day (1731 days later) »