Self-Study Set Theory for Grad Algebra Prereqs

[cap.r]

hey so I am taking my first grad class as an under grad next term and am looking to make sure i have all the prereqs. I am good with the 2 semesters of undergrad algebra and most of linear algebra although it's been a couple of years since that one.

my concern is with set theory. I have been reading stuff on wiki and they all use set theory vocab that I have little familiarity with. can you recommend a good self study set theory book that will give me what i need for taking gad algebra.

thanks

 

[rasmhop]

my concern is with set theory. I have been reading stuff on wiki and they all use set theory vocab that I have little familiarity with. can you recommend a good self study set theory book that will give me what i need for taking gad algebra.

Vocabulary such as what? Stuff like union, element, inclusion? Cartesian products? Stuff like Zorn's lemma, axiom of choice, well-ordering? Stuff like ordinals, cardinal arithmetic? Usually not that much set theory is required for grad algebra besides what you must have acquired in a year of undergrad algebra. Most students know the basic rules for working with sets (including unions and stuff like that, rules like de Moivre's laws and constructions like (finite) Cartesian products). Most grad algebra texts have either an intro or appendix discussing infinite Cartesian products, Zorn's lemma + axiom of choice, and possibly well-ordering and cardinal numbers. Try to check your grad algebra book and see if that suffices. If you're not ready for that yet, then check you undergrad algebra textbook which likely has a discussion of the more basic set theory (or check your other undergrad books, many books between calculus and grad stuff contains sections on set theory). We need to know what you're expected to know, and what you know before we can make recommendations.

One book I thought I would mention is Halmos' Naive Set Theory which is a nice little account of just about what you would need for grad algebra.

 

[eof]

He might mean category theory where you need to worry about the concept of a class. Our graduate class also expects you to be fluent with transfinite induction. We didn't cover any of that in class and were just told to read it up during the first week of class if we didn't know it.

EDIT: My school has a separate grad class for Ph.D. and Masters students. The former is a lot harder than what most top schools offer and goes into homological algebra, spectral sequences etc. pretty quickly. Almost everything is also defined categorically by adjoint functors. This makes it possible to define for example polynomial rings in a way that works also in the case of a non-commutative ring and which reduces to the usual definition in the commutative case.

We were also assumed to know cardinal arithmetic.

 

[rasmhop]

Well if he's expected to know category theory and uses a standard textbook, then likely that textbook has a section on category theory. Algebra by Hungerford, Algebra by Lang and Basic Algebra 2 by Jacobson all use some category theory (hungerford barely any, Jacobson a fair bit), but they all have sections on it. As for a category theory reference I like Handbook of categorical algebra, vol. 1 by Borceux, but it's quite expensive and I haven't heard anyone else recommend it (don't know why).

 

[cap.r]

"Cartesian products? Stuff like Zorn's lemma, axiom of choice, well-ordering? Stuff like ordinals, cardinal arithmetic?"

yeah that stuff. i hear the kids who are in that class use these all the time while talking about their homework. they were talking about transfinite induction and i looked it up on wiki. I honestly wouldn't be able to do it... the book they like to use at my school is usually abstract algebra by Dummit

 

[rasmhop]

"Cartesian products? Stuff like Zorn's lemma, axiom of choice, well-ordering? Stuff like ordinals, cardinal arithmetic?"

yeah that stuff. i hear the kids who are in that class use these all the time while talking about their homework. they were talking about transfinite induction and i looked it up on wiki. I honestly wouldn't be able to do it... the book they like to use at my school is usually abstract algebra by Dummit

Dummit-Foote has an appendix entitled "Cartesian Products and Zorn's Lemma" which gives a concise overview of this stuff. It goes over the axiom of choice, the well-ordering principle, Zorn's lemma, partially ordered sets, infinite Cartesian products.

The main text is fairly light on the set theory and while it has an appendix on category theory it's never used I believe (at least not for the first half or so). I don't know how your lectures and problems will be structured, but if you follow the book it shouldn't be a major problem to learn the required set theory. The most important thing is that you familiarize yourself with Zorn's lemma and Cartesian products as these are extremely important. If possible you should get the book early and try to read the appendix. It may be a bit too concise if you've never seen Zorn's lemma, but once you actually see it in action it should become easier to work with. I would still like to recommend Naive Set Theory as it's meant for a person at about your level I think. It goes back to basics when it comes to set theory and presents it in a more rigorous and thorough fashion than most people have had it presented. Of course being a book on set-theory and not a prep book for other courses it will take some time to get through and only a bit of it will be applicable to your grad algebra course, but it will probably give you more confidence in working with set theory which is very important in all future courses.

 

[johnshade]

The Halmos book is a good rec. His mission statement is summed up in the last few sentences of his introduction: "In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some and here it is; read it, absorb it, and forget it."