part 11 of set theory (toc)
Semester’s over, jokers. Here’s the last contenty set theory post (I might get around to uploading solutions later to some problems that I thought were interesting).
I’m just going to dump some stuff from the end of the semester to wrap up for now. It’s kind of a pity because (in my opinion) this stuff was by far the coolest stuff of the semester. Oh well.
part 9 of set theory (toc)
Let’s get cracking on ordinals. No time wasted or lost.
Enderton introduces ordinal numbers as a way to measure the “length” of a well-ordering, just as cardinal numbers were a way to measure the “size” of an arbitrary set.
part 8 of set theory (toc)
And we go a-blazin forward, on the trail of set theory catchup.
part 7 of set theory (toc)
So I submitted five graduate school applications yesterday. That $100 application fee per university hurts, man. I’d been working pretty much exclusively on my thesis up until this point, getting it presentable enough to submit with my applications. No doubt I will need to polish it more, but it was good enough to be sent in. So now I’m playing catch up with my classes, especially set theory.
I’ve basically lagged behind for two chapters, which are cardinals and ordinals. I think I will probably do at least two posts on each chapter. The notions are related, and (in very broad terms) can be thought of as two different ways of dealing with infinity. Proofsareart has a brief discussion here. Going along with this kind of parallelism, my professor structured the material so that we’d learn something about cardinals, then the similar thing about the ordinals, and so on, so we kind of straddled the two chapters at once. I appreciate him kind of suggesting the similarity this way, but I found it kind of confusing to constantly be switching paradigms, in a way. So I think I’ll just keep them separate and do cardinals first, then ordinals. Maybe if I can be bothered, I’ll remark on some links between the two along the way.
part 6 of set theory (toc)
Meant to do this post ages ago. Forgot. #sorrynotsorry
part 5 of set theory (toc)
So several people have been on my case for not posting enough. All right, you caught me. But I’ve been really overwhelmed with my imminent graduation, graduate school applications, and my research… but, hey, time’s a-wastin’ and you aren’t here to hear me complain.
So my set theory professor has a qualifying exam that every student must ace in order to pass the class. And I’m serious about acing. I forgot the exact mark you need to get, but I think it’s somewhere north of 95%. It’s pretty fair though, because we are told in advance exactly the flavour of questions we’re going to get. Well…if he’s emphasising these things so heavily, they must be important. Onto the blog they go.
part 4 for set theory (toc)
At this point I’m convinced it isn’t a math class unless you construct some sort of number system. Reals from rationals in analysis, rationals from integers in algebra, and now the natural numbers from… well, from literally nothing at all. Construction ain’t so bad, although I’ve seen a lot of hype for wrecking balls lately too…
part 3 of set theory (toc)
Super super quick thing. Interesting problem on this week’s homework.
isolatedvertex asked: The most common formulation of the axiom of choice is: for any set S, there exists a function whose domain is the set of non-empty subsets of S and such that for all X ⊂ S, f(X) ∈ X. Such a function is called a choice function for obvious reasons: for all X, it "chooses" an element of X as its value f(X). The controversy (which is much less fierce nowadays than it used to) arises only when S is infinite: then the other axioms do not provide a way to construct a choice function.
as always, thank you (:
part 2 of set theory (toc)
I literally can’t deal with set theory right now. My brain wants to explode. Here, have some set theory.
part 1 of logic (toc)
This is probably my last year as an undergrad as well (!) but I’m trying cram a lot in. Since Set Theory already has an initial post, I think I’ll start in on some of the first notes for the other math class I’m taking this semester which is Mathematical Logic. Some of the goals of this course are to:
- Provide a foundation for mathematics;
- Define a language for mathematics;
- Define a notion of mathematical truth; and
- Define what a proof is.
Lofty ambitions, but technically rigorous. (Technically correct is arguably the best kind of correct at least according to an old high school friend of mine). Anyways, we’re going to use a bit of Group Theory for some examples eventually so feel free to brush up on that as necessary [toc].
part 1 of set theory (toc)
Hello everyone, and welcome back to my final (!) year of undergrad. I’m only taking one math class this semester because of my other major, and that class is set theory. I can already tell this class is going to be really confusing if I don’t try and keep things straight from the outset.
Anyways, what better way to ease back into the whole writing-about-math thing than doing one of those elusive short and sweet posts? Here are two weird things from set theory that have already happened so far…and we’ve only had one day of class!
part 4 of complex analysis (toc)
Anyone waiting for elegant methods and results for complex analysis that were supposed to be imminent— we’re about to do some cool stuff but just a few quick refresher definitions!