Best mathematical set theory books according to redditors
We found 52 Reddit comments discussing the best mathematical set theory books. We ranked the 14 resulting products by number of redditors who mentioned them. Here are the top 20.
We found 52 Reddit comments discussing the best mathematical set theory books. We ranked the 14 resulting products by number of redditors who mentioned them. Here are the top 20.
Here's my rough list of textbook recommendations. There are a ton of Dover paperbacks that I didn't put on here, since they're not as widely used, but they are really great and really cheap.
Amazon search for Dover Books on mathematics
There's also this great list of undergraduate books in math that has become sort of famous: https://www.ocf.berkeley.edu/~abhishek/chicmath.htm
Pre-Calculus / Problem-Solving
Calculus
Linear Algebra
Differential Equations
Number Theory
Proof-Writing
Analysis
Complex Analysis
Functional Analysis
Partial Differential Equations
Higher-dimensional Calculus and Differential Geometry
Abstract Algebra
Geometry
Topology
Set Theory and Logic
Combinatorics / Discrete Math
Graph Theory
P. S., if you Google search any of the topics above, you are likely to find many resources. You can find a lot of lecture notes by searching, say, "real analysis lecture notes filetype:pdf site:.edu"
As someone just finishing their last year of Masters in maths undergrad, A lot of the stuff that you find in The Art of Problem Solving won't really show up until year 2 probably.
Here are the books I used in the summer before starting uni
"How to think like a Mathematician"
Bridging the Gap to University Mathematics
A Consise introduction to Pure Mathematics
Those books were interesting reads for me so I would recommend them. I'll answer any questions you have if you want.
I would guess that career prospects are a little worse than CS for undergrad degrees, but since my main concern is where a phd in math will take me, you should get a second opinion on that.
Something to keep in mind is that "higher" math (the kind most students start to see around junior level) is in many ways very different from the stuff before. I hated calculus and doing calculations in general, and was pursuing a math minor because I thought it might help with job prospects, but when I got to the more abstract stuff, I loved it. It's easily possible that you'll enjoy both, I'm just pointing out that enjoying one doesn't necessarily imply enjoying the other. It's also worth noting that making the transition is not easy for most of us, and that if you struggle a lot when you first have to focus a lot of time on proving things, it shouldn't be taken as a signal to give up if you enjoy the material.
This wouldn't be necessary, but if you like, here are some books on abstract math topics that are aimed towards beginners you could look into to get a basic idea of what more abstract math is like:
Different mathematicians gravitate towards different subjects, so it's not easy to predict which you would enjoy more. I'm recommending these five because they were personally helpful to me a few years ago and I've read them in full, not because I don't think anyone can suggest better. And of course, you could just jump right into coursework like how most of us start. Best of luck!
(edit: can't count and thought five was four)
I'm a theorist, so my book recommendations probably reflect that. That said, it sounds like you want to get a bit more into the theory.
As much as I love Awodey, I htink that Abstract and Concrete Categories: The Joy of Cats is just as good, and is only $21, $12 used.
Another vote for Pierce, especially Software Foundations. It's probably the best book currently around to teach dependent types, certainly the best book for Coq that has any popularity. You can even download it for free. I recommend getting the source code files and working along with them inline.
I will say that I don't think Basic Category Theory for the Working Computer Scientist is very good.
Real World Haskell is a great book on Haskell programming as a practice.
Glynn Winskel's book The Formal Semantics of Programming Languages is probably the best intro book to programming language theory, and is a staple of graduate introduction to programming languages courses.
If you can get through these, you'll be in shape to start reading papers rather than books. Oleg's papers are always a great way to blow your mind.
I totally hear that. I jumped back into math two years ago after a decade break, I thought I knew linear algebra well, but I only knew it in R2 and R3 for videogames and physics applications. I had a vague sense of what eigen values were and such, but one of the books I went through recently was Axler's 'linear algebra done right'. I was a little unprepared, haha. I made it through and learned a lot though. Here's what I've learned.
First up... there's a whole knew way of thinking ahead of you. There are two basic skills it would seem. The first is learning to find examples to illustrate a given point. For what functions does this property not hold? Why? For which does it do? A related skill, is learning to find further questions. If this property only holds with this class of functions, what happens if we look at other classes? If real symmetric matrices can always have a square root, is this ever true by happenstance with a matrix that isn't symmetric?
The trick here is to learn to stop taking things as they're given, and start pushing the boundaries. Ride the rules hard, find where they break, have specific examples in mind that you can use to help remember the properties of a given thing. I work with stats a lot, so covarience matrices are what I think about since I've spent more time with them.
As a side note: keep an eye on when your examples aren't strong enough. If you need to be thinking about anti-symmetric matrices but you only vaguely know how they work, figure out how you can get more intimately familiar with them. What kinds of practical applications do they pop up? What kinds of problems can you wrestle with for a while to bulk up your comfort with these kinds of examples?
the second skillset, is learning to think with proofs. It sounds crazy I know, but as you pick up this skillset, the proof as the explanation will start to make sense... like... you'll be pissed off if someone hand waves instead of using a proper proof format. There's a reason it's used, but it's a radically different way of thinking than we're used to. It takes time to learn new languages, so spend more time with it, it'll come.
A book I'd highly recommend... check out Alcock's 'how to think about analysis'. It's a quick read, it'll only take you a week or two, but it'll go into a lot more detail on some of this stuff, and might hopefully help you wrap your brain around what you've been missing.
The last thought is structured review. I use anki. I like to make short little flash cards to help illustrate some point I learned (maybe 'sketch a proof of this' or 'what is the definition of a normal operator in terms of how it acts on vector lengths'. Just little pieces of stuff that I understand in a moment of clarity, but that I don't want to just forget the next day, you know? Don't make long cards that will be a pain in the ass to review, or if you do to schedule regularly returning to a problem, make sure it's in a different deck so you don't throw off your other review.
Anyway... the real advice from there is to just log the hours. Ich habe auch viel Zeit damit verbracht, Sprachen zu lernen. Nicht so gut als dir, but it's given me a lot of understanding and patience for math. This isn't a bag of tricks to learn, it's a new language. It takes time for your mind to acclimate, form the patterns, and get used to the new logic and syntax. Don't beat yourself up, just start spending time with it. Learn new ways to approach abstract problems, get comfortable with thinking in terms of proofs, and get used to assembling a grab bag of example objects you can use to test the limits of a given theorem, and help you think more concretely about a problem (how would this property hold for 2x2 rotational matrices?)
beyond that, I've heard good things about A Concise Introduction to Pure Mathematics, but if you're wanting to master proofs and prepare for analysis or linear algebra, Axler's linear algebra book or Spivak's calculus might be better choices more specific to your interests, though they'll obviously have far less hand-holding and be more of a trial by fire. Either one combined with a few insights from Alcock might be enough to bootstrap your way in though.
I'd personally also make sure to always have a math book you're actually excited about that you work on in the background as a hobby. Something that's not tainted by any classes, teachers or tests... just a bizarre journey to wonderland that's just for you. Strogatz 'nonlinear dynamics and chaos' and 'visual complex analysis' have both been really interesting ones I've spent time poking around in on the side over the last few months. It's nice to have at least a corner of your math journey that's just about thinking about weird stuff, and not about deadlines and obligation and stress.
Try Naive Set Theory by Paul Halmos. I think it's aimed at undergraduates, so the content is a bit dense, but the style and tone is very conversational and engaging. I thoroughly recommend 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 have to second Dummit and Foote as a supplement to Lang's text, they're pretty much complete opposites; where Lang is very to the point (terse, some may say) and from a very abstract viewpoint, Dummit and Foote has a lot of exposition and examples and is done from, what at least what I would call, an appropriate level for a first graduate course in abstract algebra. It also has an appendix that deals with category theory, it's nothing extensive but it may help you become more familiar with the ideas of category theory. I am currently using this book for a graduate course in algebra so I have some familiarity with it; it is a bit too wordy for my tastes but that may be your thing.
A book with which I have limited experience but quite like so far is Mac Lane and Birkhoff's Algebra it's done with the same general perspective as Dummit and Foote but it has a bit more category theory (it is introduced at the end of the third chapter and the entire fifteenth chapter is dedicated to category theory), it isn't terse but it is less wordy than Dummit and Foote.
Another (very) popular choice (but one with which I have no experience) is Aluffi's Algebra: Chapter 0 it develops category theory pretty much from the start and supposedly is much less terse than Lang (I only say supposedly as I have no first hand experience with it).
If you want something that only deals with category theory, the classic text is Mac Lane's Category Theory for the Working Mathematician I have found looking at this book for a long period of time has helped me with understanding/getting used to categorical ideas. I also have experience with this book for which you can find on the internet (legally) for free and I find it rather good.
Halmos
You need some grounding in foundational topics like Propositional Logic, Proofs, Sets and Functions for higher math. If you've seen some of that in your Discrete Math class, you can jump straight into Abstract Algebra, Rigorous Linear Algebra (if you know some LA) and even Real Analysis. If thats not the case, the most expository and clearly written book on the above topics I have ever seen is Learning to Reason: An Introduction to Logic, Sets, and Relations by Nancy Rodgers.
Some user friendly books on Real Analysis:
Some user friendly books on Linear/Abstract Algebra:
Topology(even high school students can manage the first two titles):
Some transitional books:
Plus many more- just scour your local library and the internet.
Good Luck, Dude/Dudette.
I would learn some mathematical logic (Enderton, Leary or Chiswell and Hodges are the usual suggestions. I'd avoid Enderton's, personally), and then come back to Kunen or Hrabeck and Jech (pdf). If you're feeling particularly ambitious, Jech's other book (pdf) is the reference for working set theorists, so may want to take glances at that while reading Kunen or H&J.
If you'd like to read up on set theory, there are plenty of reasonably priced books available, particularly from Dover. Here is one such book. I don't own this book in particular, but Dover books are usually pretty decent given their extremely low price point, and aren't usually written to be dense as bedrock. That being said, don't expect to absorb everything through osmosis by just skimming the text. If you really want to learn and absorb the material, you will have to sit down, read, re-read, and work, but it's only an insurmountable task if you tell yourself it is. If you have any questions, or need help or insight, you can feel free to ask them here on /r/math (though they'll probably be best put in the simple questions sticky) or over at /r/learnmath.
I don't know where to even start :)
Infinity is a property. Sort of like an adjective. You don't say something is an infinity, but rather something is infinite in size. Think of "infinite" rather than "infinity".
A set is a collection of unordered objects(anything at all), like so
{shoe, car, &, 3}. It's unordered because we can write it as {car, 3, shoe, &}.
A set is said to be finite if we can pair everything in it with another reference set {1, 2, 3, ..., n}.
In the reference set "..." means that numbers continue in the given order. "n" at the end means that there's a number at which the numbers terminate.
Lets call our set {shoe, car, &, 3} A. So, A = {shoe, car, &, 3}.
Now compare the elements(objects) inside A with those inside {1, 2, 3, ..., n}:
shoe can be pared with 1.
car can be pared with 2.
& can be pared with 3.
3 can be pared with 4.
So everything in A is pareable with everything in {1, 2, 3, ..., n}.
So, A is finite according to our definition above.
Definition: S is an infinite set if and only if there exists a set A such that A is a proper subset of S and |A| = |S|.
Ok. This is one of the definitions of infinite set and to understand that you need to be familiar with the notions of functions, mapping, cardinality, bijection, equality, existence, proper subset...all pretty basic notions.
Tell you what, why don't you just study these books below that would teach you all about these notions and much, MUCH more?
A Book of Set Theory by Charles Pinter.
Naive Set Theory by Paul Halmos.
The books above will not only teach you about finite/infinite sets, but also can serve as a very nice foundation to study higher math.
IMO while Jech's is a great book, it is not a good first book on set theory. It is meant for graduate students and assumes a fairly high level of mathematical capability form the start. I'm working through the book now, and I'm still in the "basic" part but it's already covered topics like the Borel hierarchy, non-principal ultrafilters, and measurable cardinals: certainly not introductory topics.
The best introductory book I've seen is Discovering Modern Set Theory. It is very well-written and introduces all the basic concepts including ordinals, cardinals, ZFC axioms, and some basics of model theory. It also covers a lot of prerequisites like formal logic and a little abstract algebra, which the OP is likely not familiar with.
The same Pinter who wrote the much lauded and dirt cheap Dover text on introductory abstract algebra recently came out with a book on set theory, also a dirt cheap Dover text.
It looks really great and covers quite a bit.
I highly recommend this book here. Got it for my undergrad, it's concise and affordable and covers many different topics so you can flick around, with problems at the end of each section, also very affordable compared to other uni-level books. I don't know what level you are at but I think it's suitable for anyone heavily into maths, pre university/college. Very neat book
Hi,
I'm a TA in my school's CS theory course (a mixture of discrete math, and the automata, languages and complexity topics most CS theory courses cover).
As others have said, "theory" is pretty broad, so there are an awful lot of resources you could look at. As far as textbooks go, we use two - Sipser's Introduction to the Theory of Computation (which others have recommended), and the freely available textbook Mathematics for Computer Science, by Lehman, Leighton and Meyer - which concentrates more on the "discrete math" side of things. Both seem fine to me. Another discrete-math–focused set of notes is by James Aspnes (PDF here) and seems to have some good introductions to these topics.
If you feel that you're "terrible at studying for these types of courses", it might be worth stepping back a bit and trying to find some sort of an intro to university-level math that resonates for you. A few books I've recommended to people who said they were "terrible at uni-level math", but now find it quite interesting, are:
These aren't specific to computer science, but might be of help. Good luck!
Liebeck's Concise Introduction to Pure Mathematics is a great text for introducing students to the basic tools required in abstract algebra, number theory and analysis, but doesn't go into great depth.
It's kind of a standard text but for abstract algebra I think Dummit and Foote is remarkably clear.
Ireland and Rosen's Classical Introduction to Modern Number Theory is a classic, but maybe more intermediate.
Elementary Number Theory by Jones is very good.
If you’re that motivated I’d recommend studying a proper proof based university level math textbook in your spare time, most of the classes offered at high school are boring and don’t have much to do with actual mathematics.
This is a great introduction to pure mathematics: https://www.amazon.co.uk/Concise-Introduction-Mathematics-Third-Chapman/dp/1439835985
If you're completely new to the subject, I would pick up an introduction to proof, such as Velleman's How to Prove It. They all have a chapter featuring a good introduction to set theory.
If you're up for something a little bit, but not too much, more challenging, you could pick up Halmos' Naive Set Theory and learn a bit about the axioms underpinning set theory.
For a first course have a look at Goldrei's book -- his book on logic is excellent. He's a good pedagogue and doesn't shy away from the "real stuff". It might be a bit low-level for your grad students but see what you think.
Set Theory:
Naive Set Theory
Number Theory:
Elementary Number Theory
Introduction to Analytic Number Theory
A Classical Introduction to Modern Number Theory
Topology:
Topology
Introduction to Topological Manifolds
I second Hrbáček and Jech followed by Kunen for a thorough, rigorous treatment. But Set Theory and the Continuum Problem by Smullyan and Fitting is another interesting, self-contained exposition that concentrates on consistency and independence proofs, the axiom of choice, and the continuum hypothesis. It covers both Gödel's and Cohen's proofs. It says it does not have any prerequisites, but that does not make it easy. It also has interesting philosophical asides.
You can ask for a reading course from that professor.
Here's two inexpensive references 1 , 2
> I couldn't find any example of that on the web. Do you have a link to such proof?
It's in Kunen's Set Theory chapter 1.
We can prove it ourselves. Assume the axioms of extensionality, pairing, union, and specification. Let A be a set and assume the power set of A exists. Let f : A -> P(A) be any function (formally, we assume it's a set of ordered pairs satisfying the definition of a function with A as its domain and P(A) as its range). Then X = { x in A | x is not in f(x) } is a set by specification. And there is no a in A such that f(a) = X. If there were, this will lead to the usual contradiction (a is in f(a) if and only if a is not in f(a)).
At no point do we appeal to the axiom of foundation. We need the basic axioms to be able to say what it means for f to be a function, and the axiom of specification to be able to conclude that X is a set.
> I meant a universal set U that contains all possible sets within a given set theory. But it's perhaps misleading to call that a universal set.
Again, in any set theory that contains the axioms I mentioned, there is no set of all sets. It wouldn't be a set. In some sense, it's too "big" to be a set. So there is just no such thing.
Theory of Sets by E. Kamke. I bought a used copy the summer before I started college. The combination of the subject matter, his compact style of writing, and the old-school German typography really inspired me to want to learn more math.
Yeah, I love the intro to that book. The intro to his other text, A book of set theory, is a great history of the foundations of mathematics as well. I would definitely recommend either of Pinter's books for this purpose, they are self contained and nicely motivated.
> It is really hard to find something that actually is - AFAIK (I'm not very advanced in math so I could be wrong) nobody has actually found a statement that we know fits the example.
I'm not sure what you mean by "the example" here, but this doesn't sound right. Godel's own proof constructed such a statement that was independent of Russel & Whitehead's theory. The R&W stuff has fallen out of favor these days, but Godel's construction has been shown to work everywhere relevant.
The new hotness is a set theory call Zermelo-Frankel Set Theory (colloquially, ZF). And we now know, for example, that several really important hypotheses are independent of ZF (and PA):
Scott Aaronson has a very interesting survey on the state of P vs. NP that covers relevant ground:
The definitive work on Independence Proofs (afaik):
I would recommend reading further into this topic. This is a book I am reading: http://www.amazon.com/Theory-Continuum-Problem-Dover-Mathematics/dp/0486474844
Sadly, I can't think of a title in discrete math or introduction-to-proofs that I can recommend. A common recommendation for the latter category, which I haven't read myself but has a good reputation, is the following:
Another book which has a good reputation is
The book even has its own Wikipedia article!
---
These, however, are both about proofs as their own technique. I wish I could provide a recommendation for books on discrete math, introduction to set theory, and the related topics I mentioned above. You might consider something like
(This title also has its own Wikipedia article, too.)
but I'd defer to others for recommendations on textbooks for these prerequisite concepts and principles useful to an analysis student.
---
>Also, when should I start my real analysis? Can i study it with the calculus or after completing calculus?
I'd consider taking real analysis after completing the introductory sequence in calculus, possibly including multivariable calculus, linear algebra, and an introduction to differential equations. I'd also wait until after you've had a good introduction to mathematical proofs, something most universities and colleges present in a class on discrete mathematics.
If you jump into analysis completing at least one-variable differential and integral calculus, as well as a class with a strong proof-based component, you're likely to find yourself in over your head.
First, most analysis classes assume the students are already familiar with ideas like convergent sequences, limits, continuity, differentiability, and integration. This is all presented again in a much more rigorous way, but it's typical for an analysis class to lean heavily on prior interaction with such topics.
Second, analysis is a heavily proof-based class, and learning how to read, understand, and write proofs is its own skill set. Trying to acquire fluency in proofs by taking an analysis class, despite no prior formal encounters with proofs, will make analysis considerably more challenging for you.
I hope this helps some. Good luck!
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.
I haven't used the set theory books myself so I can't comment on their quality, but anytime I hear someone looking for reasonably priced math books I immediately think of the Dover Books on Mathematics series.
https://www.amazon.com/Theory-Logic-Dover-Books-Mathematics/dp/0486638294
https://www.amazon.com/Axiomatic-Theory-Dover-Books-Mathematics/dp/0486616304/ref=pd_bxgy_2/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486616304&pd_rd_r=c84f7c07-350b-4ec3-8fc5-adf30ae9b20c&pd_rd_w=74PSR&pd_rd_wg=RG95z&pf_rd_p=a2006322-0bc0-4db9-a08e-d168c18ce6f0&pf_rd_r=RKXGP4020J1B5GVED0PH&psc=1&refRID=RKXGP4020J1B5GVED0PH
https://www.amazon.com/Book-Theory-Dover-Books-Mathematics/dp/0486497089/ref=pd_sbs_14_2/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486497089&pd_rd_r=82e4d26d-281c-4eb6-982f-5811be6be764&pd_rd_w=gx29l&pd_rd_wg=O6GtQ&pf_rd_p=43281256-7633-49c8-b909-7ffd7d8cb21e&pf_rd_r=8TQ89WSVK726CHBY6N96&psc=1&refRID=8TQ89WSVK726CHBY6N96
https://www.amazon.com/Naive-Theory-Dover-Books-Mathematics/dp/0486814874/ref=pd_sbs_14_1/131-2870981-9872902?_encoding=UTF8&pd_rd_i=0486814874&pd_rd_r=82e4d26d-281c-4eb6-982f-5811be6be764&pd_rd_w=gx29l&pd_rd_wg=O6GtQ&pf_rd_p=43281256-7633-49c8-b909-7ffd7d8cb21e&pf_rd_r=8TQ89WSVK726CHBY6N96&psc=1&refRID=8TQ89WSVK726CHBY6N96
edit: added more books
Abstract and Concrete Categories is what I used. It's written at a pretty high level, but it's understandable iirc.
To piggyback of of /u/blaackholespace, if you're trying to study mathematical logic as a discipline in its own right, I would recommend Enderton's Mathematical Introduction to Logic, which covers the basics of model theory and proof theory up to the Incompleteness Theorems.
If you're looking for a deeper study of set theory (if not, that's cool too), I would recommend Kunen's Set Theory: An Introduction to Independence Proofs.
Discovering Modern Set Theory by Just and Weese is exquisite.
Hello! I'm interested in trying to cultivate a better understanding/interest/mastery of mathematics for myself. For some context:
 
To be frank, Math has always been my least favorite subject. I do love learning, and my primary interests are Animation, Literature, History, Philosophy, Politics, Ecology & Biology. (I'm a Digital Media Major with an Evolutionary Biology minor) Throughout highschool I started off in the "honors" section with Algebra I, Geometry, and Algebra II. (Although, it was a small school, most of the really "excelling" students either doubled up with Geometry early on or qualified to skip Algebra I, meaning that most of the students I was around - as per Honors English, Bio, etc - were taking Math courses a grade ahead of me, taking Algebra II while I took Geometry, Pre-Calc while I took Algebra II, and AP/BC Calc/Calc I while I took Pre-Calc)
By my senior year though, I took a level down, and took Pre-Calculus in the "advanced" level. Not the lowest, that would be "College Prep," (man, Honors, Advanced, and College Prep - those are some really condescending names lol - of course in Junior & Senior year the APs open up, so all the kids who were in Honors went on to APs, and Honors became a bit lower in standard from that point on) but since I had never been doing great in Math I decided to take it a bit easier as I focused on other things.
So my point is, throughout High School I never really grappled with Math outside of necessity for completing courses, I never did all that well (I mean, grade-wise I was fine, Cs, Bs and occasional As) and pretty much forgot much of it after I needed to.
Currently I'm a sophmore in University. For my first year I kinda skirted around taking Math, since I had never done that well & hadn't enjoyed it much, so I wound up taking Statistics second semester of freshman year. I did okay, I got a C+ which is one of my worse grades, but considering my skills in the subject was acceptable. My professor was well-meaning and helpful outside of classes, but she had a very thick accent & I was very distracted for much of that semester.
Now this semester I'm taking Applied Finite Mathematics, and am doing alright. Much of the content so far has been a retread, but that's fine for me since I forgot most of the stuff & the presentation is far better this time, it's sinking in quite a bit easier. So far we've been going over the basics of Set Theory, Probability, Permutations, and some other stuff - kinda slowly tbh.
 
Well that was quite a bit of a preamble, tl;dr I was never all that good at or interested in math. However, I want to foster a healthier engagement with mathematics and so far have found entrance points of interest in discussions on the history and philosophy of mathematics. I think I could come to a better understanding and maybe even appreciation for math if I studied it on my own in some fashion.
So I've been looking into it, and I see that Dover publishes quite a range of affordable, slightly old math textbooks. Now, considering my background, (I am probably quite rusty but somewhat secure in Elementary Algebra, and to be honest I would not trust anything I could vaguely remember from 2 years ago in "Advanced" Pre-Calculus) what would be a good book to try and read/practice with/work through to make math 1) more approachable to me, 2) get a better and more rewarding understanding by attacking the stuff on my own, and/or 3) broaden my knowledge and ability in various math subjects?
Here are some interesting ones I've found via cursory search, I've so far just been looking at Dover's selections but feel free to recommend other stuff, just keep in mind I'd have to keep a rather small budget, especially since this is really on the side (considering my course of study, I really won't have to take any more math courses):
Prelude to Mathematics
A Book of Set Theory - More relevant to my current course & have heard good things about it
Linear Algebra
Number Theory
A Book of Abstract Algebra
Basic Algebra I
Calculus: An Intuitive and Physical Approach
Probability Theory: A Concise Course
A Course on Group Theory
Elementary Functional Analysis
Check out A Concise Introduction to Pure Mathematics by Martin Liebeck. I found it a useful stop gap between uni and a levels!
I've always heard Naive Set Theory by Halmos is good. Note, that it isn't actually about naive set theory, but axiomatic set theory
http://www.amazon.com/Naive-Set-Theory-Paul-Halmos/dp/1614271313