From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition.
My question is what the definition of a set is?
I have noticed that many other definitions start with a set and then something. A group is a set with an operation, an equivalence relation is a set, a function can be considered a set, even the natural numbers can be defined as sets of other sets containing the empty set.
I understand that there is a whole area of mathematics (and philosophy?) that deals with set theory. I have looked at a book about this and I understand next to nothing.
From what little I can get, it seems a sets are "anything" that satisfies the axioms of set theory. It isn't enough to just say that a set is any collection of elements because of various paradoxes. So is it, for example, a right definition to say that a set is anything that satisfies the ZFC list of axioms?