Student Learning Set Theory Independently

[praeclarum]

Hi,

I am a high school student, and I am trying to learn set theory (it's not for school - independent study; I love it). I have a book (Takeuti/Zaring Introduction to Axiomatic Set) Theory and I am going through it, but I feel like I'm going way to slowly (after trying to go through it for 2 months now and countless hours, I'm only just completed chapter 8 on ordinal arithmetic out of 20 chapters or so. And this is like for 1.5 hours a day and the first 4 chapters you can finish in a day.). Furthermore, I feel like I don't quite understand the idea of ... say, an isomorphism. Like, I could explain to you exactly what it is, but I feel like I am just spitting out the definition, and that I don't have a true understanding of it. Plus, sometimes I get really stressed for some reason when I do it for more than 2 hours.

It's a problem because none of my teachers can really help me with it. What do you recommend?

 

[Number Nine]

Set theory is a good place to start, but a book devoted to it may not be the way to go for someone with little experience (I assume) in pure mathematics. For a good introduction to proofs, I recommend something like Gilbert's Elements of Modern Algebra (there are some very cheap used copies on amazon). It covers the basic of set theory and will walk you through everything clearly enough that you'll quickly get the hang of abstract math (plus, abstract algebra is an absolutely fundamental subject in mathematics).

 

[praeclarum]

Set theory is a good place to start, but a book devoted to it may not be the way to go for someone with little experience (I assume) in pure mathematics. For a good introduction to proofs, I recommend something like Gilbert's Elements of Modern Algebra (there are some very cheap used copies on amazon). It covers the basic of set theory and will walk you through everything clearly enough that you'll quickly get the hang of abstract math (plus, abstract algebra is an absolutely fundamental subject in mathematics).

Ah, but I'm already familiar with a fair amount already -- I went through half of Quine's "Set Theory and Its Logic" (which is a bit more basic than Introduction to Axiomatic Set Theory). Perhaps I should finish that book (since it's a lot less terse in the explanations) ... I have plenty of experience with proofs, too, except I always work them backwards instead of forwards. I have already studied (on my own), propositional and first-order logic. I only know the basic properties of rings, fields, and integral domains, etc. (the extent of my abstract-algebra knowledge) so I will check out that book you recommended...

I was more looking for some way to substitute not having a real teacher / lecturer to explain stuff. Are there online resources or anything besides Google (which often does not completely answer the question)?

 

[micromass]

I can highly recommend the book "Introduction to set theory" by Hrbacek and Jech. It's one of the best introductory books on the topic.

If you do not understand the concept of an isomorphism, then you didn't do enough abstract algebra. An isomorphism between two "structures" mean that the structures are exactly the same up to renaming. Check out "a book on abstract algebra" by Pinter. It is suitable for high school students.

 

[mal4mac]

Maybe you're just too young/inexperienced to be attempting Takeuti? Or maybe Takeuti just isn't your style? Browse as many books as you can in libraries and bookshops to see if any appeals more to you. Look up isomorphism in the indexes, see who gives the best explanation (for you). If you get too stressed out after two hours then the simple answer is: "Do less than two hours!" If you are doing all your school work than you are doing very well if you can find two extra hours a day to do set theory. I encountered 'isomorphism' in my first year at college and had no problems understanding the concept. Maybe you just need to leave it a few years and then it will be obvious. If are getting straight As in school, and are sure you 'own' school mathematics, then why not try reading some popular books? One I've just read and would highly recommend is "The Genius in My Basement: the Biography of a Happy Man" by Alexander Masters. This gives a neat introduction to *group* theory along with a superb biography of a mathematician who was doing hard stuff at a high school age! It also explores the pitfalls of doing that. The appendix gives great advice on reading books on a hard topic like group theory (equally applicable to set theory...)