Man - you know that feeling where you make a small outline because you don't want to wander on a question, but then it still turns out to be way longer than expected?
I just made my longest answer yet - perhaps even Arturo (whose answers I highly respect - it is magical how often he manages to give long, thorough answers before anyone else answers) would think it is too long.
@mixedmath Quite so. Although I wish they had a chat related query. I have no idea how close I am to chat related badges, and from time to time I actually find interest in that.
@Jonas: I think they're equivalent, but I prefer furthermore. It's always an adverb, and I don't like guessing parts of speech (further can sometimes be an adjective, and thank goodness you didn't ask about moreover ;p)
Which is essentially to give a summarize of the proof Magidor's covering lemma based on Magidor's proof for Jensen's covering lemma. It's not hard, nor long. I'm just so exhausted that I stare at the Texmaker vacantly and do nothing.
I thought about the mathematician, who tried to prove that all cardinalities are equal, but ended up misreading Bernstein's thesis and proving something most useful.
I've been spoiled by MathJax; I was answering this question: programmers.stackexchange.com/questions/101163 and found myself thinking "golly it would be nice if this place had MathJax"...
if anyones here, I was wondering if someone would be kind enough to suggest to me a good set theory text (for a relative beginner). They don't seem to offer many (if any) courses on the subject that I can take in school but I want to get at least a decent amount of it down before studying some other subjects too in depth. Thanks
@Deven My experiences with set theory have been a bit limited - I either picked it up in some analysis class, topology class, or somewhere else related to set theory
but if you sent a message to Konig Asaf, I'm sure he'd love your distraction.
Make sure to ask for one with the AC covered in it - he'll appreciate that.
One of the guys in my office is researching in alg. graph theory, perhaps he knows about it. Although it might not be the case, since I have no idea what graph entropy is.
Well, I guess it was somewhat exciting at first, but now that I'm writing about it it rapidly stopped being that exciting
Asaf: There's some chance he might have heard it. It's used in some algorithms, as far as I know
I'm writing a note in which I expose three definitions of graph entropy, and prove their equivalence. The only proof I have of one equivalence is the paper that originally defined the concept of graph entropy, and it's rather hard to read. It even defers some details of the proof to an impossible-to-find hungarian journal...
My question for him would be: "Is there a better and comprehensive treatment of the subject, or should I continue hunting for papers?"
You use $ to start and end the environment, and type math formulae inside. $$ would start it in a centered newline, and allow limits to be written nicely (actually above/below the symbols)
At worse case you can always type it "badly" and leave a comment request for help on the markup.
@Explorer I found the question you are referring to, and the display looks okay on my end. Which operating system are you running, and which browser are you using?
A function that satisfies both $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2$ for all real $x$ is known to be the identity over $\mathbb Q$, but is it also the identity over $\mathbb R$? If not, can you provide me an example of such a function? Thanks.
@AsafKaragila Oh, why are they working on two major version numbers ahead of the current released one? That's like against all reasonable software development practices I've heard about.
@AsafKaragila The Chrome model is already plenty stupid in my opinion. The fact that they are spending energy on an alpha release when the previous version is still in beta just makes me even more pessimistic about it.
@WillieWong Clearly. However the Mozilla developers seems to have felt as though they have smaller genitalia (code-wise, that is) and have to release faster to appease the audience.
@Explorer: if you are actually using a Nightly build, then unfortunately I can only tell you that Beta and Alpha softwares are not supported by StackExchange, and I think neither by MathJax (the engine that renders the mathematics), since tweaks to the Javascript engine and MathML engine may be responsible for the breakage, and those could be bugs on the side of Firefox, and not "our problem"
@Explorer Not really. These numbers are means of control, for the end user to know how far into development things have gone. Firefox <4 had slow paced development which gave a very good sense.
Since 4.0 changing major version number becomes an idle change, so to speak.
@Explorer You are really running firefox 7 or 8? Does that work on other websites heavily using Javascript? (Just out of curiosity. I am way too old to live on the bleeding edge like that.)
@Explorer Mathematically, sure. Feasibly? Not really. I am not sure how much programming experience you have, but you seem to misunderstand the idea of version numbering.
@AsafKaragila You know that I run Gentoo, right? Also, I think that whole thing about 3.0.0 is just a whimsical thing, and not a change of the whole development model.
This is just like serial numbers for software are not "randomly generated numbers", but the result of (often too simple and stupid) algorithm. The version numbering is too to indicate the amount of change from the previous version.
@Explorer Because with enough programming experience you understand better what is a big change, and what is a small fix. Therefore what is the difference between 5, 5.0.1 and 5.1
(Answered for future reference)
It looks like the browser is caching a wrong version of MathJax. Try to clear the browser cache by holding Shift and click Refresh.
If that still doesn't work, please post a bug report (tagging as (bug) and (tex)) on meta.math.stackexchange.com
@Explorer I'm not trying to make you seem smaller or bigger. I'm just saying, there is, or at least supposed to be, a difference. If you don't notice it - it's a whole other story.
@WillieWong I asked on meta. Apparently I am the most active jerk around here... :)
@Explorer Have you tried Willie's solution, to clear up the cache and try again? If javascript works fine on other sites then it should be a cache problem. Otherwise perhaps a JS problem.
@AsafKaragila Well, I am pretty sure since he was just here at Cambridge on the MIT-Cambridge exchange program, which is open for third year students in MIT and second year students in Cambridge. (Because undergrad is 3 years long in UK and 4 in US.) That and the fact that he is a SE intern now. Graduate students tend not to work in internships.
@Explorer that is just a bit over 10 percent, right? Also, in a lot of the cases the questions are either not answerable, or just doesn't have upvotes.
(Not answerable can either mean that the question is poor enough that it doesn't have any reasonable mathematical answers, or that the question is so good that it is actually an active research topic, so no one knows the answer.)
Just like a neurologist might not be a leading expert on nephrology.
@Explorer That's a stupid reason. I think that epistemology is exactly the proof that it's very easy to ask but almost impossible to actually answer.
Also, that question itself requires a very technical answer, which would require some background before that. It is about six pages long without definitions and previously needed theorems in a rather concise book I have on the topic.
@AsafKaragila In fact, for most mathematicians, the corpus of knowledge that he/she is familiar with is pitifully small compared to the entirety of known mathematics.
@AsafKaragila That's not right. I distinctly remember GEB having a countable many number of pages. The only book to have omega pages is the Book of Sand
"The Book of Sand" (original Spanish title: "El libro de arena") is a 1975 short story by Jorge Luis Borges. It has parallels to "The Zahir", continuing the theme of self-reference and attempting to abandon the terribly infinite.
The story appears in a book of the same name, the Spanish language version of which was first published in 1975. The English translation by Norman Thomas di Giovanni was first published in The New Yorker; the entire volume The Book of Sand (ISBN 0-525-47540-0) was published in 1977.
Plot summary
The titular "Book of Sand" is the Book of all Books, and is a m...
@Explorer Shame, probably the best series ever to be shown on TV. In one of the episode the protagonist (Number Six) is facing a supercomputer which supposedly can answer any question.
So to speak, if it can be asked - it can be answered.
Number Six destroys the computer by overloading it, by feeding it a most simple question that has absolutely no complete answer.
@Explorer Basically, mathematical proofs are based on logic. Unfortunately, we cannot get something out of nothing. So there always have to be certain things that we assumed to be "obviously true". Godel's incompleteness theorem states that given a sufficiently powerful system of logic (in the sense that it contains enough assumptions to make mathematics "meaningful") that is not self contradictory (that you cannot deduce that a statement is simultaneously true and false), then (cont.)
(cont.) there exists a statement that can be stated in the framework of this logic system but cannot be demonstrated to be either true or false in this logic system.
Actually, when I went to visit Brown, I sat next to him in a grad number theory class (he has taken many a grad class - embarrassingly, he could take my quals and do better than me like right now)
@mixedmath Yeah, I knew Zev was an undergrad. I visited his page... However Qiaochu shows an extensive knowledge and capability of explaining it. Even more than the average grad student. So it's dissonant with the fact he is an undergrad :)
But hey, despite my coarse and abrasive exterior I am a fun person when you get passed that. It just takes a while to get through... usually some alcohol is needed as well.
@Explorer: unfortunately, I looked around and cannot find an obvious answer to your MathJax problem. One possibility is to try it on another browser and see if it works (Opera, Chrome, or IE). While it is unlikely it could be a network problem. (Math.SE is a MathJax partner, so the MathJax code is loded directly from MathJax content distribution network.)
Oh, I am fully aware of my unimportance. In fact this is what drives my arrogance. No one is important. Nothing is important. This is why I find great comfort in set theory and infinities. It has no actual importance, as these are not real objects.
What do you say when people ask you why you do set theory?
My parents ask me why I do math all the time, and they expect me to respond in ways that make it dazzling, important, and exciting to everyday life. Like I'm supposed to be a writer for NUMB3RS or one of those shows.
And why do you think I'm teasing you? Why would you assume so much about your importance in my boredom relief at this moment? Weren't you the one to complain about how people always assume they are more important than they really are?
Outspoken - hmm. I don't know that I've seen 10 users on chat... ever. But if anyone, it would have to be our great Chat Leader, Konig Asaf, I suppose.