You may have noticed me using asking many questions on Infinite Cardinalities on this fine website. Although many of the answers to my questions here were very in-depth and amazing, I just can't help but feel helpless as these questions are really tough for me.
Although I love to ask questions on Math.Stackexchange, I don't want to flood it with elementary infinite cardinality questions.
So does anyone know of any good websites or books that provide rigorous yet very in-depth proofs for those of us who are not at all mathematically inclined? My textbook is mostly informal proofs that are based on pictures. I know my professor won't like that very much!
Thank you!!
EDIT: The book I am using at the moment is Mathematical Proofs" by Gary Chartrand.
My favorite reference on set theory is Kaplansky's Set Theory and Metric Spaces. I bought this as a university freshman. I am now a research mathematician*, and I have never needed to know more set theory than that text contains.
I also wrote up several handouts on very elementary set theory that some people have told me they find helpful. To find them, go to my expositions page and scroll down towards the bottom. I should say that, following after Kaplansky but not following after several of the most set-theoretically interested and knowledgeable people on this site, the exposition is "naive" rather than "formal" in the technical sense: I do not discuss the axioms of ZF(C) or any other set theory. Also the Axiom of Choice is not discussed until the third handout, whereas in the earlier handouts some results which are not theorems of ZF set theory (i.e., without the Axiom of Choice) are nevertheless happily proved, e.g. that every infinite set admits a countable subset is an easy result from this naive perspective: since a set is infinite, you can keep removing elements one at a time and you never exhaust the set.
I do feel that this level of "naivete" is appropriate for a general mathematical audience, but if you want to be more formal in your approach to set theory you will need to look elsewhere. (Even still, I might suggest that you spend at least a little bit of time with the naive perspective before going on to study an axiomatic approach. Going from literally no set-theoretic skills to axiomatic set theory sounds tremendously confusing to me.)
*: I have a paper in which the main result is proved by (well-ordering a set and then using) transfinite induction. I was so thrilled that I ran around my department telling everyone I could find. Typical reaction: "That's great! By the way, what's transfinite induction?" Later Bjorn Poonen explained to me how to remove the appeal to the Axiom of Choice, which was honestly a little deflating.