Hey, small question. What is the group operation in Hom(A, Z_m) where A is an abelian group with exponent m? Is it addition of homomorphisms? I thought it was just function composition at first, but that can't be right.
Random fact: if Y is a group object, then Hom(-, Y) is also a group object... in the functor category! This partly explains why Hom(X, Y) is a group when Y is abelian.
There we have concepts like "first authors", if you use natbib it will show mainly the first author, while in mathematics we place them in alphabetic order.
I just dislike the [1] and so in my references. I much prefer to have a year and names or something like that. I dislike, as well, the abbreviation of the writer.
With apalike2 you have a nice reference, for example [Jech, 1973] and so.
Yes, I find it quite interesting that sheaves are now strongly associated with algebraic geometry (as I commented in my question a few weeks ago).
Hmmm, I just read through the answers in the ‘Does it ever make sense not to go to the most prestigious graduate school you can get into?’ question and I'm confused that no-one seems to say anything about the importance of choosing a department that actually does research in your areas of interest... :-|
Well now, I'm going to resume my actual work. Now that I have a few days without assignment I can focus on side projects related to the axiom of choice.
@JonasTeuwen No, they recommend either alphabetical or plain. None of the apa/nat/etc etc
Yes. I don't like their styles because they decapitalize titles. I was raised on the tradition that titles have capital letters in the beginning of most words.
@JonasTeuwen My only reason for not liking [1] citations is that it annoys me a bit to have to flip back and forth between the text and the bibliography. At least if the author is mentioned, I've more or less an inkling what's being talked about.
I'm wondering whether mathematicians know the citations for famous papers and books off by heart. Then you could say something like Wiles [1995] and everyone will know what you're talking about...
@JM That's not what the reflection in the mirror said. In fact, everything is about me! Including PDE, sadly. When countries go to war it is about me. Each side wants the other to take me... :P
Think about it, if you answer something nontrivial, the amount of people who can understand the answer and upvote is small. In fact, the amount of people who will even bother to check the question is small.
Asking something this basic is something most people can answer, so most people check it out, and most people upvote Qiaochu's simple answer because it's direct ans simple.
The amount of work I had put into some set theoretic question was immense. Except for one answer that had 20+ votes, most of them stabilize around 10 votes or so. If you go back to answers I wrote six months ago, there were less people interested and five upvotes would be a nice average.
Qiaochu's short and to the point answer got only +4, my slightly longer and approaching differently got +4 and Arturo's wonderful answer which was so educational and important got also +4.
Why? Well, it was only viewed 137 times, which considering the fact four people were involved in the question means that it is likely that much much less than 25 people even saw the question.
@Asaf: Random question: What are the problems associated with the ‘set’ of all groups? I mean, it's not hard to prove that the collection of all groups is a proper class, but I'm looking for a ‘reason’ more like Russell's paradox.
And it happens that as a consequence of this axiomatisation, the collection of all groups is a proper class. But perhaps there's another ‘reasonable’ axiomatisation of set theory which permits such sets?
Ah. I never gave such thought to the idea. The point is that it can be easily corrected within set theory to have a set for which "it is enough" to work with, or create a sequence of sets defined with a parameter so you can write a theorem covering all these sets.
For example, clearly every singleton is a trivial group. So you take isomorphism classes, and Scott's trick should be enough to allow you to have a set equivalence class. We can then limit the cardinality of groups we want to work "All finite groups" for example. And we can later prove that for every \kappa the theorem holds for "All groups which has cardinality less than \kappa".
Outside set theory I don't recall many people working with anything more than P(P(P(N))) in cardinality, so I don't think there's an actual set theoretic problem for most people.
When I first started studying large cardinals, I always felt that assuming something like inaccessible is weakening a theorem or things like that.
After spending a full year seeing, studying and reading about large cardinals... well, weakly compact (which is much much much larger than mere inaccessible) is still very small and very minor as an added assumption.
The point about these things is that once you know a trick that translates the problematic statement into a valid assertion, you can keep your problematic statement as it was and sleep peacefully.
I will let you in on a very important understanding that I had due to a chat with some PhD student in Jerusalem once.
The point that ZFC+Inaccessible proves the consistency of ZFC is not a problem at all. The theory ZFC+Inaccessible is a stronger theory, and just like ZFC proves PA this stronger theory proves ZFC.
I can't really articulate my suspicions about the inaccessible cardinals. I mean, on one level, I'm thinking, they can't be reached, so how can they exist?
But on the other hand there are many more mundane things that can't be named, and I would like to believe those exist.
I have a question, not about the mathematics. Is it acceptable to incorporate the historical stories into the questions posted? Here I mean the history of the questions, and some related topics. Is it true that, as I was afraid of, people are not in the mood to watch the stories, not directly related to the mathematics, or, what I am worried about the most, people are not concerned about it here?
@awllower I'd say it greatly depends on the length.
If it's no more than two paragraphs then I would consider this as a possible addition. If it is longer I'd think twice. At any case be sure to mark the relevant question at the beginning or the end in a bold font so people could find it easily.
I'm too tired to read the whole thing... but the thing that makes me suspicious is that it looks too short. :p Or maybe I missed a few tricks when I learned the proof.
Oh wait, you're not proving that it forms a long exact sequence? Hmmm. Then it looks too long. :-/
But this time all the content is directly relevant!!It talks about the backgrounds, and does not exceed the required amount to explain it. I hope this can turn out solved, as I already spent a deal of time on improving it; moreover, after some brave but strange tries, I have had a more coherent understanding of the question, and posted it already.
Also, I received one suggestion that I shall use the Latex; does this mean that I shall improve upon the typing skills? If so, I have to ask questions about it to do so...Is this some kind of the foundations of mathematics..Haha!!
@awllower Using LaTeX is a good idea for two reasons:
1. It makes your post vastly more readable and pleasant to the reader;
2. It gives the impression that you are trying to be serious about your writing.
While the second is a very subjective reason, it is in my opinion, a very prominent boost to your post. When reading posts (esp. long posts) if I see that the writer is formatting his question well, I will usually consider it more serious than if the writer did not format his question well.
Oh, I cannot agree with you more. And I often use it when appropriate. What I mean is that, is there other ways of showing you are serious? If not, then it probably should be included as a necessary tool for the students, which I hope, but found no teacher!!
Of course, for the dearth of right understanding of the latex, I from time to time use some skills in writing to replace this function, which apparently did not work fine. Or, per chance, my English is not good enough to use this skill!!
@AsafKaragila : I would try to format the question as well as possible, thanks for your suggestions.
@awllower You should. As for the LaTeX? Well, I can honestly say (and as far as I know this goes for many mathematicians) that when you open up a paper which was not written in LaTeX you immediately get suspicious about the validity of its contents.
@awllower: The way I like to think about it is "my potential audience's time is valuable; I will do all that I can so that reading my question is easy on their eyes..."
@AsafKaragila, and @JM : Thank you all for the suggestions, as one of my mentors once said, to do mathematics ought not to be hurry. I guess I was somewhat in a rush!! I should be able to fix this.
But, on the other hand, I still know nothing to help, when the question is indeed too long, and needs some backgrounds...
Just a thought: if the question requires way too much setup, maybe ask some professor in the nearest academic institution to help you with formulation at the very least?
Ah indeed, I should do so, or just ask the professors? In any case, thank you very much; now I know why not so many people were interested in the question. I have to sleep now, and have to u=figure out why the functions and their expansions do not agree with each other, on some curves, which is what I came up with, and thought of as the real problem of my post.
Ok, then there is nothing I can do here to beg for help^^ I think I had better study the algebraic geometry even harder, so as to understand the key...
That would be quite an impressive feat, given I knew nothing to begin with!
I don't think even reading Linderholm's Mathematics made difficult could have that sort of negative impact on understanding. But perhaps a sufficiently malicious author could write something...
Is it possible for someone with little set-theoretic knowledge (me) to understand the proofs that CH or AC is independent of ZFC?
I am looking for any kind of mathemtical sounding statement (Not "This sentence is unprovable") for which the proof of independence is somewhat accessible.
It is difficult. I am going to give a very general outline, using actually the proof by ZFA permutation models - which is a much simpler method, but has the drawback of not producing ZF models, but rather ZFA.
I am writing the definitions because I will need them anyway, and it is simple enough to "cross breed" the proofs, especially if you don't get all the technical details in order.
math.stackexchange.com/questions/58318/… Why was this question closed? It falls very much within the domain of this site and has infact even been tagged appropriately as physics and mathematical physics.
@SivaramAmbikasaran The same thing can be said on many statistics related questions that we still migrate over to stats.SE or cstheory things. I feel that this question is not strongly related to the site, and as the OP said he asked it on physics.SE as well, I see no reason to keep it open here too.
@ZhenLin I did forget to tell you that the Levy collapse is the simplest form of forcing there is. Even simpler than Cohen forcing for adding more real numbers, which too is fairly simple.
Ah. I was going to remark that there's about 20 pages in Sheaves in Geometry and Logic about the use of Cohen forcing to falsify the continuum hypothesis.
No one can count more than 3, it's just impossible.
The forcing itself is very simple once you have the definitions at hand, you do require one additional lemma (which is why in the question I chose the Levy collapse). Cohen forcing does not collapse any cardinals. If x was a cardinal in V it is the same cardinal in V[G].
My advisor got a book with a proof of the independence of CH based on topos. He once told me that he never really got around to read it, and if I want to read it with him. I smiled and said that not really... as I had enough to read about large cardinals and forcing.
@AlexeiAverchenko If you are interested in number theory (or want to find out if you would be) then yes. It covers a lot of the tools (and frustrations) you'd spending time with.
Of course, it's fine to read things for fun as well, but I assume that you're at the point where you have to husband your time.
The one really nice thing about Serre is that if he makes something seem hard, then it is actually hard. That's a rare quality.
I'm reading about an example of inverse limits, and I had another small question. Let {H_i} be a sequence of normal subgroups in G, with H_{n+1}\subset H_n. Take f:G/H_n\to G/H_{n-1} to be the canonical homomorphism...what is the canonical homomorphism here? I'm used to seeing a map G into G/H, with x in G mapping to [x], the equivalence class in G/H. But what would a class [x] in G/H_n map to in G/H_{n-1}?
My gut feel about "If something is valuable to the site, it should generate reputation" would be "a cautionary tale could be nice and useful, though I'm not comfortable about getting rep for that..."
@yunone Suppose you have F < H < G, F and H normal in G. Then we can define f: G/F \to G/H as f(xF) = xH. Proof of correctness: let xF = yF. Then y \in xH, and thus xH = yH. Proof of homomorphism: f(xyF) = xyH =xHyH = f(xF)f(yF), f(F) = H. You can think about his homomorphism as induced by the embedding F \to H. (I hope I didn't screw up anywhere ^_^)
note that it's surjective, too, corresponding to the idea that since F is smaller than H, then G/F must be larger than G/H
Introduction to the answer:
I began writing, hoping to have a relatively short answer. However as I originally expected this is not a simple proof at all. I wrote a short guide to the construction of symmetric extensions and permutation models - the two canonical ways of proving independence fro...
Yeah. In the process I came upon a severe mistake in the book from which I was taking the proof. I had to spend an hour trying to figure out how to correct this.
Is it possible for someone with little set-theoretic knowledge (me) to understand the proofs that CH or AC is independent of ZFC?
I am looking for any kind of mathemtical sounding statement (Not "This sentence is unprovable") for which the proof of independence is somewhat accessible.
