Not sure who you mean. I remember the name Belyakin in connection with something similar, but I do not remember exactly what he tried to prove (and how it was accepted).
To sum his post, "Oh, inaccessible cardinals are most likely true. I'm actually showing that ZF (and other contemporary set theories) is not adequate for mathematics..."
@robjohn Set theory was fathered by Georg Cantor, which was a wee bit crazy himself. So it has bad genes to begin with. ZF was a topic of research for Kurt Goedel as well, which was a raving lunatic... so this cannot be a coincidence :-P
Main theorem (ZF): There are no weakly inaccessible cardinals. The proof of this theorem was derived as a result of using the subinaccessible cardinal apparatus which the author has worked out since 1976, the preliminarily investigations were developed since 1973
@HenningMakholm Well, much like the existence of an infinite set cannot be asserted without its own axiom, there is no reason to assume that the existence of anything stronger can be just... provable.
But there is still a gap from not provable to inconsistent. Currently, modulo Kiselev's work, this gap may exist or that the contradiction is still waiting to be found.
My instinct was to retag it as "calculus", but now I'm in doubt. "Calculus" is an abbreviation for "the part of real analysis that is usually taught in high school", right?
Okay, "high school" is not completely well defined. Point is "calculus" seems to be a subset of "real analysis". I can imagine something being "real analysis" because it is too advanced for "calculus" (e.g., power series). Can it also be "real analysis" rather than "calculus" simply because it involves neither a derivative nor an integral?
@HenningMakholm To muddy the waters even more... at my university we teach two semesters of "advanced calculus," which, according to my colleagues, is more advanced (duh) than calculus but not as advanced as real analysis.
Advanced calculus course description: "This course is an introduction to advanced analysis. Topics of study include set theory, the topology of Euclidean spaces, functions, continuity, differentiability of functions and mappings, integration, series, uniform convergence, transformation of multiple integrals, differential geometry of curves and surfaces, and vector calculus."
I haven't taught this course yet so I haven't quite figured out how this differs from real analysis.
Sounds like an "overview-of-everything which should be adequate as a general knowledge basis for subjects where you elect not to take the dedicated course" course.
We don't actually teach a course called "real analysis." Whatever real analysis our students get comes from this course. (We only teach undergraduates here.)
@tb It depends on the institution. And, as I said, I haven't quite figured out how "advanced calculus" differs from real analysis, other than it's supposed to be less advanced.
It's my impression that U.S. undergrad math programs (except, I guess, at the best schools, although maybe not even there) aren't as advanced as a lot of the undergrad math programs in, say, Europe. I can't back that up much, other than that it's also my impression that U.S. colleges spend time teaching topics that are often taught in high school in Europe.
Thus, for instance, math majors here often have to take a lot of writing and humanities courses in college, whereas (again, my impression) students in Europe spend more of undergrad focusing on their major subject.
@tb Yes, no measure theory or functional analysis in "advanced calculus" is one of the major distinctions.
@Gortaur That's also true. We don't do much theory in the first calculus sequence, so it gets shoved into advanced calculus.
@Gortaur A large part of the problem is that the high schools in the U.S. aren't all that great. Few of the students we get are ready for theory in the first calculus sequence.
@MikeSpivey In my time the usual rule-of-thumb was that the last year of Danish high school corresponded to the freshman year of an American college. So the level at the end of our 3-year bachelor degrees would be roughly equivalent to a 4-year American degree.
@MikeSpivey I don't know about all of Europe but judging from what I know about central Europe studying at universities is pretty much confined to the specialtie(s): except at "technical universities" where you focus on one subject right away you usually have a main subject and one or two second subjects (like math/physics or history/germanistics+pohilosophy chemistry/biology) for example. General culture and writing are trained marginally at best unless you focus on them.
@Mike: my colleague is from Poland and he told me that there they have in the first semester general topology to get rid of those who isn't abstract enough. I would prefer starting with metric spaces though - quite general and still easy to motivate young students.
@Alex: do they really introduced obligatory theology in secondary schools by the way?
Thanks to the Europeans for chiming in on their experiences. Virtually all universities in the U.S. have breadth requirements (literature, history, fine arts, basic math and science) that account for most of the first year. I suppose that lines up with Henning's description of how Danes viewed their system vis a vis the U.S.
@Gortaur In defense of christianity.SE.... I have visited there a couple of times and was surprised at how civil the discussions were, given the emotional nature of the subject matter.
I wonder how they can have any questions that are not immediately closed as "will likely solicit opinion, debate, arguments, polling, or extended discussion"?
@10k+ users: I see an undelete vote here. I don't like when people delete their questions after getting hints/answers, so I'm tempted to add mine. What do you think?
We can now design bacterial genome from scratch, as mentioned in this thread on skeptics.se, and this opens up new and exciting possibilities for solving many problems the humanity is facing today, from efficient generation of drugs and fuels to pollution and world hunger (although the latter mig...
God cursed Cain to be an eternal wanderer, so he went to a (probably figure-of-speech) land of wandering, the land of Nod, this much I understand. But then he builds the first city, Enoch. How is it compatible with being an eternal wanderer?
UPD: The principal source on which I base the claim th...
does it matter what education i got beforehand? what are the conditions for receiving financial aid? how much living there would cost? these kinds of questions, they are just not explained very well
what are the academic prerequisites?
what kind of research is expected from me?
i'm clueless :(
how am i going to write the statement of purpose if i don't have a coherent idea of what i'm going to study and research?
do i have do research before i even get a chance at applying?
Stanford has an online application system for grad school. You can find all the admission information on their website. Make sure your writing is good, in particular you will have to write a motivational letter. Publications will increase your chances. And most importantly you need a good score in the GRE test, both subject and general. The latter should not be a problem, maybe except for the part with the vocabulary test.
'hi, i studied as an applied math undergrad and then pure math grad in a school in a small town in Siberia, but then i realized i'll have to study again because that one was worthless. i'm interested in ag, at, ct, ha, and potentially qft.' something like that?
@Clash What's wrong with pharmine's and Aryabhata's answers? The former seems to contain a detailed proof while the latter contains an excellent outline. AM-GM is this by the way.
@Clash If x_n \geq 0 then the inequality x_{n+1} - \sqrt{a} \geq [1/x_n times (something)^2] shows that x_n+1 \geq \sqrt{a}, right? in particular x_{n+1} \geq 0. This is the induction step.
@Clash Oh, sorry it was an equality and I missed a factor of 2. I just took the first and last part of the equality after "we note that" in pharmine's answer.
Reading The Shape of Inner Space, I've come to a realization. The difference between a layperson and a mathematician at heart, is that the former gets bored seeing math in science books and the latter quietly thinks to themself while reading pop-sci books, "This is getting boring; where are the equations?"
@t.b.: Good. I hope you don't think I'm a suitor now. No worries about writing back soon, I just needed to know whether it got there or not.
@anon! Hey, how are you? You got me thinking. It really bothers me that there are clever people out there, with education, who have to pawn their laptop : (
Eh, alright. Like I said I put all my files on an external so this is just temporary. (In theory.) The bigger problem is the Wii, whose past games and hours of save data I have to get back :)
Sent out some applications, will send out a lot more tomorrow cause I'm free. Friend who owes me 150 says he'll try and get it today/tomorrow and I get paid tomorrow too (but not much). The only code I every knew was BASIC a la TI calculators and Microsoft Visual, but I've long forgotten all of that. Teaching myself properly would require a laptop :/
@Matt: True, but they pay better than minimum. Anyway, I may be a bit idealistic. I was once tutoring the son of a CEO back home and got hella money that way... heh.
You'll have to be clever about selecting potential employers. Look for ones who have an intensive interview process. My last employer had me doing tests and exercises the whole afternoon before giving me an offer...
Bye for now, I'm not getting anything done while logged into this chat. Good luck @anon! And @Clash: I think getting the privilege of private tutoring and then having dinner right in the middle doesn't show much appreciation for other peoples' time.
