If I read a book about analysis or set theory or algebra etc I start with an introductory course (obviously) but if math, at the end is just a bunch of theorems and proofs, what is the difference between what I study and what someone who's more advanced studies? More theorems? How "deep" can you go into a subject? For example, if I learn 5 introductory books perfectly about set theory, how much am I missing from the "whole picture?
In my experience, after the introductory [usually, survey] books, you'd need to become comfortable with the material in at least three more modern textbooks before you can make sense of recent research articles in that field.
For instance, while I can't speak for set theory, if you wanted to read an article in algebra, you would read an introductory text in group/ring/field theory, then you might pick up a book in commutative ring theory, then you might pick up books in homological algebra and algebraic geometry, and depending on the research article, it might now be accessible to you.
Of course, maybe not, so then you'd need to pick up a book on number fields over elliptic curves and another book on Iwasawa theory, and then you're in trouble, because once the community has given up on giving a theory novel names and instead just names the theory after one of its developers, you know you're rather deep.