TITLE: Different types of Set Theory
QUESTION [10 upvotes]: I have just now learned that there are different types of Set Theory. I read Naive Set Theory by Paul R. Halmos, but other than that I have no other knowledge of other...set theories.  
Could anyone explain the difference between Naive Set Theory and Axiomatic Set Theory? I thought they were the same thing, but apparently that is not true. Or possibly provide sources to research this topic? 
What about NBG and ZF set theory? 
I originally thought that Naive Set Theory was simply the topic of Set Theory, but for introductory level. It relies on axioms and such. How then is this different than axiomatic set theory?

REPLY [2 votes]: Naive set theory is enough set theory to get by in most of mathematics, but suffers from a number of unfortunate paradoxes.
In naive set theory, we call any collection of elements a "set." We don't say what the elements are, nor do we place any restrictions on which elements can come together to form a set.
However, this simple view has some problems, one of the most famous of which is Russel's Paradox:

Does the set of all sets (call it $S$) which do not contain themselves, contain itself?

If $S\in S$, then $S \not\in S$, and vice versa: a contradiction.
Axiomatic foundations of set theory (the most common of which is ZFC) resolve this and other paradoxes by providing a rigid set of axioms, and specify, for example, what the "elements" of sets are. In ZFC, in fact, elements of sets are just other sets. ZFC also poses restrictions on what kind of sets you can form, preventing things like Russel's paradox from happening.