Reddit mentions: The best mathematical set theory books

the 'theoretical' roadmap I think is arguably the easier one to organize. There might be faster ways to get a rough idea, but if you want the BS foundation, you'll basically want to work up to Bishop's pattern recognition and machine learning, and elements of statistical learning. Applied predictive modeling is a great practical book, that one alone could get you enough to be dangerous on Kaggle, but the deep understanding will come as you push up into the other two.

Bishop's is the easier of the two, but it'll still take a pretty mature grasp of statistics, proof based mathematics, multivariable calculus, and even a little bit of calculus of variations. How's your linear algebra? I've just started poking into Boyd's intro book just for shits and giggles (sometimes it's fun to go back and play the early levels in a game you've mastered, haha) and I'm liking the organization and topics quite a bit, I think it'd be a solid one to learn from the first time. Strang's Linear algebra book would also be a really good choice... you'll learn a little more from Strang's, but see far fewer meaningful applications that are relevant to what you're wanting to head towards. I've heard Strang's is good for calculus too if you still need that as well.

I like Wasserman's 'All of Statistics', but that book's no joke. If you aren't comfortable with proof based mathematics yet, you'll need to start there. Alcock's 'how to think about analysis' is an incredible primer, you can blow through it in a week and you'll learn a ton (if you haven't already taken a course in real analysis). If you need a primer on proof based math, this one looks like a good pick but I haven't gone through it myself yet, so I'm a little less up on that one.

Oh, you don't need combinatorics for stats (you can muddle your way through without it) but it will come up. If you really want to make sure you have rock solid fundamentals, 'a walkthrough combinatorics' is a really incredible book. It's a strange one, the chapters are fairly short and easy, and then there's always like 20 really bizarre problems. Not like 'apply what you've learned in a mechanical way' but like, REAL math puzzles. It'll be a discouraging book if you just want to blow through, but if you want something to push you to think crazy things you've never thought before, that's a great place to pick up some combinatorics and practice your proof skills.

I've been through a good chunk of that list so far over the last two years, and while there's still an absolutely hilarious amount to learn, I'm starting to see things as they are, it's pretty cool. A lot of ML architectural choices now feel very well motivated (what's softmax? Where does the cross entropy loss function come from? Why might the KL divergence be a good choice for unsupervised classification? Why is the Wasserstein metric an improvement for GAN stability?) but... Christ there's a lot to learn, haha. So if you're going to do tackle the theory, buckle up for the long haul, and make sure you're ALSO consistently making headway on the applied side. Code every day, always have something cool you're excited to be building. Work through fast.ai, or implement your own ML library. Hell, use CUDA if you like, or do a project to scrape your own data or... you know. There's a million things to do, make sure you don't wait until you're done with your math before you start doing cool stuff. Getting in over your head and doing stuff that you don't fully understand can be a great way to learn too. Plenty of times I've had math stuff 'click' because it grounds something I did a year or two back and didn't 'get' fully at the time. t-SNE and so on...

Anyway. Good luck. It's a long journey, but in my experience so far at least, if you know your shit there's opportunity out there for work too. But you'll need to really dig in to differentiate yourself, and you'll need to get good at networking. Getting a first job without a degree will put you at a disadvantage, but it's a disadvantage that'll disappear by your first two or three years working in the industry it seems like. Experience talks.

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

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:

• theoretical computer science (essentially a math text)
  
  
• set theory
  
  
• linear algebra
  
  
• algebra
  
  
• predicate calculus
  
  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)

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:

• How to Prove It: A Structured Approach, 2nd edition by Daniel J. Velleman.
  
  Another book which has a good reputation is
  
  
• How to Solve It: A New Aspect of Mathematical Method by George Pólya.
  
  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
  
  
• Naive Set Theory by Paul Halmos
  
  (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!

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.

Oh, I'm terrible at calculus, haha. I teach discrete maths and logic, and never have to touch calculus at all, thank goodness :)

But a younger friend of mine is doing calculus just now, so I'll find out what he found useful and PM you. He did say that some of the books I'd recommended him were immensely useful for maths generally (not necessarily calculus in particular). In roughly ascending order of difficulty:

• Gowers, Mathematics: A Very Short Introduction (<http://amzn.to/1TQNEWY>)
  
• Devlin, The Language of Mathematics (<https://www.amazon.com/Language-Mathematics-Making-Invisible-Visible/dp/0805072543>)
  
• Stewart, Concepts of Modern Mathematics (<http://bit.ly/1L7rY4i>)
  
• Liebeck, A Concise Introduction to Pure Mathematics (<http://amzn.com/1439835985>)
  
  So you could pick one of those, and see if it meets the level you're after. There are also some standard books on "how to write proofs" which often get recommended: Velleman, How to Prove It: A Structured Approach <https://www.amazon.com/How-Prove-Structured-Daniel-Velleman-ebook/dp/B009XBOBL6>, and
  Solow, How to Read and Do Proofs: An Introduction to Mathematical Thought Processes <https://www.amazon.com/How-Read-Proofs-Introduction-Mathematical/dp/1118164024>, and they might be useful. I've never read either, shamefully, tho I probably should.
  
  Also: something that can be handy once you're past the basics is the amazing (but large and expensive) Princeton Companion to Mathematics. Which has something useful and interesting to say about pretty much every major area of modern mathematics.
  
  I hope that helps!

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.

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:

1. Understanding Analysis by Steve Abbot
   
   
2. Yet Another Introduction to Analysis by Victor Bryant
   
   
3. Elementary Analysis: The Theory of Calculus by Kenneth Ross
   
   
4. Real Mathematical Analysis by Charles Pugh
   
   
5. A Primer of Real Functions by Ralph Boas
   
   
6. A Radical Approach to Real Analysis by David Bressoud
   
   
7. The Way of Analysis by Robert Strichartz
   
   
8. Foundations of Analysis by Edmund Landau
   
   
9. A Problem Book in Real Analysis by Asuman Aksoy and Mohamed Khamzi
   
   
10. Calculus by Spivak
    
    
11. Real Analysis: A Constructive Approach by Mark Bridger
    
    
12. Differential and Integral Calculus by Richard Courant, Edward McShane, Sam Sloan and Marvin Greenberg
    
    
13. You can find tons more if you search the internet. There are more superstars of advanced Calculus like Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra by Tom Apostol, Advanced Calculus by Shlomo Sternberg and Lynn Loomis... there are also more down to earth titles like Limits, Limits Everywhere:The Tools of Mathematical Analysis by david Appelbaum, Analysis: A Gateway to Understanding Mathematics by Sean Dineen...I just dont have time to list them all.
    
    Some user friendly books on Linear/Abstract Algebra:
    
    
14. A Book of Abstract Algebra by Charles Pinter
    
    
15. Matrix Analysis and Applied Linear Algebra Book and Solutions Manual by Carl Meyer
    
    
16. Groups and Their Graphs by Israel Grossman and Wilhelm Magnus
    
    
17. Linear Algebra Done Wrong by Sergei Treil-FREE
    
    
18. Elements of Algebra: Geometry, Numbers, Equations by John Stilwell
    
    Topology(even high school students can manage the first two titles):
    
    
19. Intuitive Topology by V.V. Prasolov
    
    
20. First Concepts of Topology by William G. Chinn, N. E. Steenrod and George H. Buehler
    
    
21. Topology Without Tears by Sydney Morris- FREE
    
    
22. Elementary Topology by O. Ya. Viro, O. A. Ivanov, N. Yu. Netsvetaev and and V. M. Kharlamov
    
    Some transitional books:
    
    
23. Tools of the Trade by Paul Sally
    
    
24. A Concise Introduction to Pure Mathematics by Martin Liebeck
    
    
25. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics by Kevin Houston
    
    
26. Introductory Mathematics: Algebra and Analysis by Geoffrey Smith
    
    
27. Elements of Logic via Numbers and Sets by D.L Johnson
    
    Plus many more- just scour your local library and the internet.
    
    Good Luck, Dude/Dudette.

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.

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.

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.

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.

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.

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

• The Art of Problem Solving - Lehoczky/Ruczyk - in two volumes, both have full solutions available separately
  
  Calculus
  
  
• Calculus - Stewart - ignore the bad reviews, it's very clear and covers all of basic calculus well - it's just the book that most intro students use and as a consequence gets a bad rap
  
  
• Calculus - Spivak - this will take your understanding of calculus to the next level
  
  
• Calculus - Apostol - there are two volumes and they are very comprehensive, albeit difficult
  
  Linear Algebra
  
  
• Linear Algebra - Lay
  
  
• Linear Algebra Done Right - Axler
  
  Differential Equations
  
  
• Elementary Differential Equations - Boyce/DePrima
  
  
• Ordinary Differential Equations - Tenenbaum
  
  Number Theory
  
  
• Number Theory - Pommersheim for a change of pace and proof skills
  
  Proof-Writing
  
  
• Transition to Higher Mathematics - Dumas/McCarthy - helps to bridge the gap between "lower" and higher math
  
  Analysis
  
  
• The Way of Analysis - Strichartz
  
• Principles of Mathematical Analysis - Rudin
  
• Real and Complex Analysis - Rudin
  
  Complex Analysis
  
  
• Complex Analysis - Ahlfors
  
  Functional Analysis
  
  
• Functional Analysis - Rudin
  
  
• Introductory Functional Analysis with Applications - Kreyszig
  
  Partial Differential Equations
  
  
• Partial Differential Equations for Scientists and Engineers - Farlow
  
  Higher-dimensional Calculus and Differential Geometry
  
  
• Calculus on Manifolds - Spivak - for a modern take on multivariable calculus
  
  
• Differential Geometry - Kreyszig
  
  Abstract Algebra
  
  
• A First Course in Abstract Algebra - Fraleigh
  
• Abtract Algebra - Dummit and Foote
  
  Geometry
  
  
• The Elements - Euclid - for culture and for understanding where math has come from, plus it's still an awesome book after all these years; don't need to go through all of it
  
• Euclidean and Non-Euclidean Geometries - Greenberg
  
  Topology
  
  
• Introduction to Topology - Mendelson
  
  
• Topology - Munkres
  
  Set Theory and Logic
  
  
• Introduction to Set Theory - Hrbacek/Jech
  
  
• Elements of Set Theory - Enderton
  
  
• Mathematical Logic - Kleene
  
  
• Set Theory and the Continuum Hypothesis - Cohen
  
  Combinatorics / Discrete Math
  
  
• Combinatorics - Cameron
  
  
• Concrete Math - Graham/Knuth/Patashnik
  
  
• Discrete Math - Biggs
  
  Graph Theory
  
  
• A First Course in Graph Theory - Chartrand/Zhang
  
  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"
  

> 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):

• The Continuum Hypothesis cannot be proved in ZF
  
• The Axiom of Choice cannot be proved in ZF
  
  Scott Aaronson has a very interesting survey on the state of P vs. NP that covers relevant ground:
  
  
• http://www.scottaaronson.com/papers/pnp.pdf
  
  The definitive work on Independence Proofs (afaik):
  
  
• http://www.amazon.com/Introduction-Independence-Studies-Foundations-Mathematics/dp/0444868399
  

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.

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

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.

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 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

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.

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.

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.

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.

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

Abstract and Concrete Categories is what I used. It's written at a pretty high level, but it's understandable iirc.

Check out A Concise Introduction to Pure Mathematics by Martin Liebeck. I found it a useful stop gap between uni and a levels!

Definitely Halmos' Naive Set Theory. Despite its name, it's an introduction to axiomatic set theory. It's just a little more naive than, say, a grad class on the subject would be.

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

You can ask for a reading course from that professor.

Here's two inexpensive references 1 , 2

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

Halmos

Discovering Modern Set Theory by Just and Weese is exquisite.

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 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.