Set theory is the mathematical theory of sets. In naive set theory, sets are introduced and understood using what is taken to be the self-evident concept of sets as collections of objects considered as a whole. In axiomatic set theory, the concepts of sets and set membership are defined indirectly by first postulating certain axioms which specify their properties. In this conception, sets and set membership are fundamental concepts like point and line in Euclidean geometry, and are not themselves directly defined.
Today, when mathematicians talk about set theory as a field, they usually mean axiomatic set theory. Informal applications of set theory in other fields are referred to as applications of naive set theory, but usually are understood to be justifiable in terms of an axiomatic system (normally the Zermelo–Fraenkel set theory).
Proofs are arranged in the following sections: