Reddit mentions of Set Theory

If we look at the "set" of all Cardinal Numbers, then that "set" can be indexed by the "set" of all Ordinal Numbers, call it V. Ordinal numbers are a standard way to talk about Well-Ordered sets. But it turns out that V has all the properties of a well ordered set, so it is a well ordered set and, additionally, it is strictly bigger than every other ordinal. But we can make the ordinal V+1, since V+1 != V we have V+1<V, but by definition of the successor ordinal V<V+1 which is a contradiction, V cannot be a set. Therefore, because the set of all ordinals cannot exist and the set of all cardinals is bijective with all the ordinals, it follows that the set of all cardinals cannot exist. Also, this does not depend on Choice or the GCH, if you don't assume them, then there are even more Cardinals so the result still holds.

This essentially says that the number of all infinites is too big for any infinity to count! Crazy, huh? Read about it here. Also if you want to start to learn more Set Theory and things I recommend looking for Suppes' book. It's a nice introduction to these ideas and will get you acquainted with the language and how things work. Once you have a bigger vocabulary and a bit more mathematical maturity I recommend Jech's book, it is gigantic, dense and tough but has everything that someone who isn't actively doing Set Theory research could ever need about Set Theory in it, if you can get through it...

When I took a graduate set theory course, the book used was Kunen's Set Theory (Amazon), which I enjoyed. I've also read through some parts of Jech's Set Theory (Amazon, SpringerLink) and liked what I read.

I'll second the Halmos text, it's short and sweet; if you're looking for something more comprehensive I'd suggest Set Theory by Thomas Jech.

I mean I've never seen the axiom of determinacy come up, but the axiom of choice comes up all over the place.

I guess it terms of textbooks on set theory if you're looking for something grad level Jech's Book is the classic, although rather expensive. Kunen also has a similar book. On the lower end there's stuff like Halmos' Naive Set Theory or Jech has a book purely on AC.

I guess what I was getting at with depth, is what do you want to know about them? Are you interested in forcing and independence proofs, i.e. showing that AC is independent from ZF? Or are you just curious about what sorts of things require AC?

Set theory. http://www.amazon.com/Set-Theory-Thomas-Jech/dp/3540440852

I did out all the proofs that are left out in the first few chapters and developed an undergraduate level paper on ordinal numbers.

If you need an introductory text into Set Theory and Logic, you should try Kunen's Set Theory: An Introduction to Independence Proofs or Jech's Set Theory.

Then I would recommend reading Aczel's paper on Non-well-founded sets (1988).

For some historical context, I would urge you to read the amazing graphic novel Logicomix.

All of these books can be found online.

>Spivak's book is a nightmare if you haven't learned much about topology or manifolds before

Hah! Same thing happened to me. I'm surprised this turned you off to research though. You do know that math has many, MANY different fields very different from differential geometry, right? Though I must say I can't blame any confusion on you; your classes so far have been a little focused... Your two classes next semester are vastly different from what you've seen before, so that's a plus.

Over the summer it might be good for you to work on expanding your horizons rather than going deeper into something you've learned. Look at graph theory, combinatorics, probability theory (did you do this in measure theory?), or statistics. How about learning a little set theory? Do you know of operator theory?

Or maybe you might be interested in some of the less popular subjects. Pick up a book on continued fractions like the ones by Olds or Khinchin. Or you could take a look at orthogonal polynomials.

Hell, it would be super useful to just learn a programming language well over the summer. Python is great for math stuff and very versatile otherwise.