In algebra, there are axioms for groups, rings, and fields. In each case, certain concepts are accepted as intuitively understood, and certain facts about them are assumed as intuitively true. It is the framework of theoretical mathematics. The most delicate and difficult part of it is proof, and Mathematical Structures generally devotes a fair amount of time to developing both its skill and its art. The concept of a set is one of the most primitive in mathematics.
Sets are important, useful, and simple.
The point is that they are the clay from which anything mathematical can be fashioned. Constructing number systems . The importance and usefulness of sets is demonstrated by showing that all our number systems can be built from sets. We can choose the empty set, or , to represent 0. For 1, we can choose , a set with exactly one element.