Prove that there are 'intermediate' or 'advanced' good set theory books out there.
I'm just kidding. But I am currently an upper division math student who is looking for a challenging book on set theory that does not assume much preliminary knowledge about set theory. NOTE ADMINS: I realize that there are already some threads on INTRODUCTORY set theory books, but I am specifically looking for a book that is challenging rather than introductory. Hence please do not mark this as duplicate. I am not looking for a book that is easy but I am looking for a book that ideally has these qualities:
(1) Explains stuff well; that is, the book gives an 'intuitive' explanation on various concepts as well as mathematically rigorous explanation of them;
(2) Has hard problems;
(3) preferably has little stuff about applied mathematics (I'm not interested in that);
(4) attempts to connect to further topics outside of the book, e.g. connection to algebra, etc; and
(5) Do not assume the reader to have preliminary knowledge on set theory except basic stuff.
I of course have basic knowledge on set theory that any math major would know.
Also, I would appreciate detailed comments about your recommended book (why you like it, etc.)