What's the differences between naive and axiomatic set theory? I'm at a loss studying math.Recently I decided to begin with set theory as it seems the most fundamental for math.I found the book Naive Set Theory by Halmos,and began to read it because it's so thin and maybe easier.I do now know what "naive" means,considered maybe basic ?
Later I searched the internet ,and found naive set theory is not axiomatic set theory,it's defined informally and even may contain contradictions.That startled me a little,what we study now is not necessarily rigorous and strict?In another book for set theory ,the author listed the axioms of ZFC at the very beginning which is not so easy for me to digest at once,but the book also illustrates the axioms ,like extension,specification,and I didn't found any differences yet.
Then which to learn?This book is enough or recommend me another?(what about Introduction to Set Theory by M.Dekker?)
 A: The title of Halmos's book is a bit misleading.  He goes through developing basic axiomatic set theory but in a naive way.  There are no contradictions in his book, and depending on your background that may be a good place to start.  Halmos will still develop all the axioms of ZFC in his book, but they will be presented in natural language and a much slower pace than most axiomatic set theory books.  If you are looking for something a bit more advanced, I would recommend either Set Theory by Ken Kunen or Set Theory by Thomas Jech.
The other thing is that set theory has a close relationship with mathematical logic, and so to understand the basics of set theory there is usually an assumed knowledge of some basic mathematical logic.  For a mathematical logic book, I would recommend Mathematical Logic by Ebbinghaus and Flum or Introduction to Mathematical Logic by Enderton.
Either way, I think Naive Set Theory by Halmos should be a good beginning point.  It is much shorter than the other books and does not require as much in the beginning.
