It's a good thing he worked with Specker's method by Quine atoms ($x=\{x\}$ sets), this allows me to generalize his results into ZF in a very smooth way while actually generalizing his result and not just redoing his proof with forcing.
Hey guys, I'm working on a homework problem...Don't really think it's worth posting a full question for it. Can you help me out? I need to calculate the Galois group of the polynomial $x^6-1$ over Q. I think I have an answer, but I'm not so sure that it's right...
A theorem is stated, and then a proof is given in which some general considerations are made so that the proof is concise....but the proof may be easy to go through just for the purpose of knowing that the theorem indeed is correct...
I am not understanding. what do you mean by "making general considerations"? I am sorry to ask but I neither make sense of it nor see what that could possibly mean!
But my question is even before start proving the theorem, can we ask ourselves, why such a thing (theorem) is necessary and and what condition are required for it, and then start deriving the result rather than just assuming the theorem is correct and start proving it......My suspicion is that the author would have thought about this way but he gives a proof which is readable and concise by making some considerations without explaining why he considered them !
The thought processes that a mathematician would have to go through to reach each theorem in turn, in a standard text such as Rudin, would make it an enormous book.
Yeah, I'm thinking back to my time at Uni. The professor would write some theorem and its proof on the blackboard, but as they talked through it, they would explain and justify these considerations.
@Rajesh: You could either be referring to "how we could discover this theorem in the wild (i.e. adventuring without knowing it beforehand)" or you could be referring to the motivation for various pieces of theory.
I believe there was an MSE question on why they always consider open sets in differential geometry. IIRC it's just that considering them in particular can be done WLOG and everything is easier this way.
@RajeshD I think I can shed some light here: Math is taught very well in my place here. We try to analyse more carefully why something comes up before attempting a proof.
why the condition of $f'(x)$ being continuous at $x = a$ and others ...Once I read the proof i figured out everything but not at the first glance....I had to work backwards....but wouldn't be benificial the proof itself was written backwards ? Do authors follow these thing with the purpose of giving some excersise to the readers
I would prefer to consider such things AFTER understanding the proof. So once the proof has been presented, you could add "... we need $f'(x)$ to be continuous at $x=a$ because ..." followed by the reason for it.
I mean if the proof relies on some condition, then it ought to be obvious how the proof breaks down when the condition is not met. Of course, that doesn't necessarily imply that the result is false in such a case.
I think academic mathematicians tend to look down on texts that explain things, rather than presenting things.
Also, everyone has a different way of learning and therefore suits a different style of explanation. Therefore, by including any style of explanation in a textbook, the author automatically limits the size of the audience.
I am reading a wrong book then.....but sometimes on some subjects, I feel good to read reference type of books, sending a lot of time on each page trying to understand more and more from what is not written in it
It's concise in the sense that there are no explicatory remarks anywhere, apart from a brief introduction to each chapter. It's really just a list of axioms, definitions, theorems and proofs.
@Kannappan : the one i was saying is syntactically akin to this (greatly simplified) though : Prove that there exists an integer $x$ such that $x+3 = 0$, now the proof in Rudin's book is akin to : Consider the integer $-3$, substitute in the given equation to see that it holds, Hence proved !
If you see the intermediate value property, that says, if two functional values are attained, all the values inbetween will be attained too. (Loose, and imprecise, may be!)
@N3buchadnezzar Why?
So, they start with a $y$ inbetween the functional value, then consider the supremum over all $x$ that satisy $f(x) \le y$.
Then they claim that at this value of $x$, the function will attain that $y$. If you look at the graph, this claim is absolutely natural. Don't you think so?
@KannappanSampath Its rather obvious, showing nearly no effort, making heaps of elementary errors, trying to solve advanced questions before understanding the basics, not thinking over the help he got, and so on.
When I took a third year physics class (must have been 1990), one assignment question was to derive some equation from three other equations. Something about magnetic fields; I don't remember the details. I messed around for a bit and convinced myself that no amount of mathematics could turn the three equations into the one that was required. So I did some fudging, where I munted all the equations together, then took a +ve square root on one side and a -ve square root on the other side.
I then multiplied, added and subtracted the right things from both sides until I had the identity I wanted. The physicist marking the assignment gave me a big red tick and a perfect score. It was embarrassing.
@DavidWallace If you write (a - ?) (5a + ?) you will not find a combination of factors that produce the desired linear term. Only (a + ?) (5a -?) works to test possibilities.
@JonasTeuwen They should... Otherwise your health insurance might cover (part of) it if a doctor confirms carpal tunnel syndrome or whatever it is you have there (it's been a while that you complain about it).
I want to ask a question about inscribing a circle between $x^n$ and $x^(1/n)$ given that $x \in [0,1]$. I am looking for a function that gives me the largest area of the circle, with a given n>1.
I imagine myself wearing that keyboard on my hands or belly or something while I have a screen display projected onto sunglasses I'm wearing. Hella cyberpunk.
@tb pretty good. I am proctoring a midterm at UCLA today, so I probably won't be doing much on the site. I just broke 18K and a shiny new silver badge today :-)
@robjohn oh, then you're reduced to teddy bears standards on main :s Oh, Sportsmanship is one of my favorite badges because it supports the vote early and often paradigm :)
@robjohn: I read in the transcript that you were trying to construct a norm on the Schwartz space. There's no single norm inducing the topology of the Schwartz space (the norm you wrote down might not be finite). You can produce a metric by setting $$d(f,g) = \sum 2^{-n} \frac{\|f-g\|}{1+\|f-g\|}$$ but that's the best you can do.
@KannappanSampath Just two major comments. First, in the proof of Theorem 1.3 you’ve labelled the two directions wrong: the first paragraph proves $(\Leftarrow)$, and the second proves $(\Rightarrow)$.
Then in the Notation bit near the top of page 3, you have $\operatorname{cl}_Y(E)=\overline{E\cap Y}$; it should be $\operatorname{cl}_X(E)\cap Y$ or $\overline{E}\cap Y$.
@robjohn I guessed so :) (I hate it when I'm noticing a glitch in a formula, try to correct it and get the "it's too late to edit that message" message)
I prefer the $\operatorname{cl}_X E$ notation to the $\overline{E}$ notation, since it makes absolutely clear in which space the closure is being taken.
Oh, in case you’ve forgotten: you haven’t completed the proof of (e) in Theorem 2.6.
@BrianMScott I will stick to this from the place where such a mistake can potentially occur. For instance, when I prove that fact about separation in $X$ iff in $Y$.
@Gigili I didn’t reject it, but that one with the sine of $10°$ wasn’t a good edit: you lost the degree sign, which really is needed, and the post was so short that there wasn’t any real need to set off the question with a >.
@KannappanSampath I mostly just checked them for correctness. There are places where I’d say or arrange things a little differently, but I don’t offhand recall anything that stood out as being especially awkward.
@KannappanSampath I am female, as you can see. And I'm referring to the whole community. You better to not think about something you have no information about.
@Gigili I thought about improving your edit by reinserting the degree sign, but then I decided that no edit was really necessary and left it to someone else to make a decision.
@Gigili: That edit of mine was in response to the comments of the question - it showed the user how to frame a textbookesque question and sent the message to others that editing the OP's post is more constructive than whining about tone. Plus I fixed a typo and added a very relevant tag, and nothing I did detracted from the quality of the post otherwise.
@Gigili Improving is not as easy as it could be. When I tried it, I lost all the original edits that the first editor made. Perhaps I did something wrong, but I have not "Improved" an edit since.
@Gigili The MSE community is overall one of the nicest and best-behaved communities that I’ve encountered on the web. Of course, I speak as a Usenet veteran, and some Usenet communities are notoriously snarky.
@BrianMScott Oh, that's OK. Thanks for looking at in the first place. I'll however have to disturb after a few more weeks when there is a substantial amount I have added. But, yes, you may tell me no as well. ;-)
@robjohn I’ve improved several, but there’s a trick to it: either you must have very clearly in mind exactly what you intend to do, or you should work in a separate tab.
@tb I’m betting that it’s the campaign against Elsevier.
Now coming to The Cost of Knowledge, I know you did not sign up, So I wanted to bring to your notice this thing which you may be interested in. As Brian already pointed out, it is the Campaign against Elsevier. @tb
@Gigili It’s not a big deal, but I do think that it’s useful to future users of the site to try to make sure that significant words in titles are spelled correctly.
@KannappanSampath Could you provide me with a link? I only seem to find that one which I didn't support because at that point I had two papers in Elsevier's printing presses and I would have found it rather hypocritical to support the campaign...
You definitely have a point, Gigili. People with enough rep are left to their own devices when it comes to editing, while those below the threshold are subjected to peer review. Who knows how many edits from the former category would actually pass through the latter.
