Two numbers walk into a set of brackets...

February 19, 2010 1:48 PM

Looking to teach myself the basics (and perhaps the not-so-basics) of set theory... How do I do it?

I'm interested in learning about set theory. I'm not a mathematician, nor do I desire to be one, but the concepts of set theory that I have a limited familiarity with are very interesting to me as ontological arguments. I've always been pretty good at math, but I haven't studied it in any real way since about 1994 when I graduated from high school. So, here goes:

1) What basic mathematical concepts/skills do I need to (re)learn? I'd say that right now, I have a decent base up through basic algebra and basic geometry. What else do I need to learn?

2) What are the foundational texts or issues, pre-Cantor, that I should read?

3) What are the key foundational texts of modern (aka, 19th C and later) set theory that I should read/study?

4) What are the important issues in post-Cantor/contemporary set theory, and what are some good overviews of these subjects?

User-friendly texts/websites for auto-didacts and non-specialists are much appreciated. I'm not looking for super-dense texts that only an expert would understand, but I also don't want dumbed-down froo-froo chicken soup for the set theorists soul crap. I'm looking to understand the concepts, the major debates, and so forth. Thanks!

posted by Saxon Kane to Education (12 answers total)

We used Halmos ("Naive Set Theory") in our Foundations class. It's a thin book, but take your time. And do the exercises. If your algebra is solid you should have no trouble.
posted by phliar at 1:59 PM on February 19, 2010

I was just staring today at the cover of a book I have on set theory, that I've never opened. For you, I flipped through it. It came to me via a very senior faculty member in my department. It's the 1964 edition of Introduction to the Theory of Sets by Joseph Breuer. It looks well laid out, with brief sections for each topic and exercises at the end of sections. Fairly readable, from my glance at it.
posted by knile at 2:02 PM on February 19, 2010

Set theory as a math subject is kind of a strange area.. most mathematicians kind of sweep it under the rug, because it tends to get in the way of 'useful' stuff. I wouldn't focus on reading original texts - you can, but its often very frustrating because the notation & vocabulary haven't quit settled down yet. The Joy of Sets by Keith Devlin is a good introduction to axiomatic set theory & most of the issues that you'd run into. It's not overly dense, and probably doesn't require much in the way of prereqs.

Set theory is the foundation for modern math, so you don't really need concepts or skills outside of it. There's this thing called "mathematical maturity" which a lot of books refer to as a prereq.. it basically means "can follow logic & (maybe produce) proofs, have seen things from a lot of different areas and can connect stuff together." But you have to start somewhere. I'd see set theory is a kind of pedantic place to start, but if that's what you're into, go for it.
posted by devilsbrigade at 2:04 PM on February 19, 2010

Best answer: I am currently working through two reference texts:

- Classic Set Theory (For Independent Study) by Goldrei

- Introduction to Set Theory by Karel Hrbacek

I recommend both. The former, as the title suggests, assumes you're reading this independently. Both assume you've studied or taken a course in real analysis.

I did a fairly thorough lit review of calculus several years ago and was surprised by how much I learned from reading the original texts, so I'll be following this closely. I doubt, however, there's as much a historical narrative to set theory as there was to calculus.
posted by geoff. at 2:05 PM on February 19, 2010

Oh, and if you're going to do this, axiomatic set theory is what you want. Naive set theory is useful as a tool, but axiomatic set theory is the only way to deal with issues actually in set theory.
posted by devilsbrigade at 2:05 PM on February 19, 2010

One way to start would be with the set theory section of just about any discrete math text. The ones I use in my classes when I teach naive (i.e. Cantor-ish) set theory are specific to my field so I won't recommend any particular one, but just about any would work to start I think. Halmos is a wonderful book, though it veers quickly into difficult territory (I'd say it is for your question 3). (Note: on preview, I want to point out that despite the title, this book is not really "naive" in any sense, and in fact is basically a presentation of ZF axiomatic set theory. I also disagree a little bit with devilsbrigade and think you definitely want to start with true naive set theory and work your way towards axiomatic set theory, which is by no means an easy topic. Axiomatic set theory may be what you want but don't start there.) It also seems like your interests would be served by reading the entry on set theory in the stanford encyclopedia of philosophy.
posted by advil at 2:15 PM on February 19, 2010

1) What basic mathematical concepts/skills do I need to (re)learn? I'd say that right now, I have a decent base up through basic algebra and basic geometry. What else do I need to learn?

Nothing. If you're interested in set theory just for its own sake, you don't need anything else. Set theory is, more or less, the foundation for everything else.
posted by number9dream at 3:33 PM on February 19, 2010

This OCW course covers the absolute basics of discrete math, including about a week of set theory. It's extremely basic, but the notes are very clear and it will be useful if you need to learn the notation.
posted by martinX's bellbottoms at 3:46 PM on February 19, 2010

I was going to recommend the Halmos book as well.
posted by leahwrenn at 5:45 PM on February 19, 2010

Response by poster: Thanks for all the information and recommendations. A follow-up question to devilsbrigade (or anyone else): What's the difference between naive & axiomatic set theory?
posted by Saxon Kane at 8:26 AM on February 20, 2010

Naive set theory corresponds to how people actually use sets in practice. It basically relies on the idea that a set is a collection of stuff, and collections of stuff behave more or less like you'd expect them to behave. The problem is, you get paradoxes (Russell's paradox being the big one people care about). So then you go to an axiomatic system where these sorts of paradoxes are resolved, at the expense of clarity of the system you're working in (very few people actually rely on ZF axioms in proofs - if you don't have to be careful about really big sets, naive set theory does just fine in most cases). And then you start adding some other axioms in and you get new problems - if you accept the axiom of choice, you get stuff like the Banach-Tarski paradox, which isn't quite a paradox in my mind but definitely doesn't agree with what should be right.
posted by devilsbrigade at 10:43 AM on February 20, 2010

What basic mathematical concepts/skills do I need to (re)learn? Spending a short while on an overview of basic proof techniques wouldn't go wrong. Start here.
posted by anaelith at 5:16 AM on February 21, 2010