I would like to request a book recommendation for axiomatic set theory that can be read after Kleene's "Introduction to Metamathematics". I like the style of "Introduction to Metamathematics" so similar style would be preferred. I would like this book to be as rigorous as possible. Due to the fact that I would like to work in the field of axiomatizing physical theories, I would like this book to contain not so many advanced topics but more information about different axiomatizations of set theory, considering their advantages, disadvantages, differences, and intuitions behind them. Of course, it may well be possible that such a book does not exist but then maybe it is possible to gain such knowledge from several books, which is perfectly fine.
I would appreciate any pieces of advice and suggestions.