Regarding set theory, we may start with some elucidatory comments.
Set theory is the mathematical theory of well-determined collections, called sets, of objects that are called members, or elements, of the set.
Pure set theory deals exclusively with sets, so the only sets under consideration are those whose members are also sets.
We have to stress the fact that it is a mathematical theory.
It is also a very simple theory, from a conceptual point of view, because its "ontology" presupposes only the existence of sets and the description of their "mathematical behaviour" is made entirely in terms of only one relation: that of "being a member of" ($\in$).
The notion of set is so simple that it is usually introduced informally, and regarded as self-evident. In set theory, however, as is usual in mathematics, sets are given axiomatically, so their existence and basic properties are postulated by the appropriate axioms.
The axioms of set theory imply the existence of a set-theoretic universe so rich that all mathematical objects can be construed as sets. Also, the formal language of pure set theory allows one to formalize all mathematical notions and arguments.
Thus, set theory has become the standard foundation for mathematics, as every mathematical object can be viewed as a set, and every theorem of mathematics can be logically deduced from the axioms of set theory.
From this point on, we have to start with the exposition of the mathematical theory itself, and with the statement of some basic principles derived from our intuition of what it means:
to "collect" into our thought some distinct objects and manage them as a single, well defined object [freely derived from the well-known Cantor's quote (Beiträge, I, 1895)].
