Set Theory is the mathematical science of the infinite. It studies properties of sets, abstract objects that pervade the whole of modern mathematics. The language of set theory, in its simplicity, is sufficiently universal to formalize all mathematical concepts and thus set theory, along with Predicate Calculus, constitutes the true Foundations of Mathematics. As a mathematical theory, Set Theory possesses a rich internal structure, and its methods serve as a powerful tool for applications in many other fields of Mathematics.
The objects of study of Set Theory are sets. As sets are fundamental objects that can be used to define all other concepts in mathematics, they are not defined in terms of more fundamental concepts. Rather, sets are introduced either informally, and are understood as something self-evident, or, as is now standard in modern mathematics, axiomatically, and their properties are postulated by the appropriate formal axioms.
A set is a well defined collection of objects. The term well defined means that the set is described in such a way that we can determine whether or not any given object belongs to that set.
Cantor moved from number theory to papers on trigonometric series. These papers contain Cantor's first ideas on set theory and also important results on irrational numbers. Dedekind was working independently on irrational numbers and Dedekind published Continuity and irrational numbers.
In 1874 Cantor published an article in Crelle's Journal which marks the birth of set theory. A follow-up paper was submitted by Cantor to Crelle's Journal in 1878 but already set theory was becoming the centre of controversy.
The history of set theory is rather different from the history of most other areas of mathematics. For most areas a long process can usually be traced in which ideas evolve until an ultimate flash of inspiration, often by a number of mathematicians almost simultaneously, produces a discovery of major importance. Set theory however is rather different. It is the creation of one person, Georg Cantor.
In the naive set theory originated by Cantor the concept of set was not defined. In particular no attention was paid to the nature of elements of sets. Since a set is understood to be a collection of objects, it was assumed that any object can be a member of a set.
The concept of set is fundamental to mathematics and computer science. Everything mathematical starts with sets. For example, relationships between two objects are represented as a set of ordered pairs of objects, the concept of ordered pair is defined using sets, natural numbers, which are the basis of other numbers, are also defined using sets, the concept of function, being a special type of relation, is based on sets, and graphs and digraphs consisting of lines and points are described as an ordered pair of sets.
Fundamental to set theory is the notion of membership - sets have members, also called elements. To express the relation of membership, we use a stylized epsilon symbol.
Infinite sets are basic tools of modern mathematics and the essence of set theory. No contradiction resulting from their use has ever been discovered in spite of the enormous body of research founded in them.
The historical importance of this role for set theory is unquestionable: the axiom of choice was the subject of controversy among mathematicians throughout the first half of the 20th century. But once again it is at least debatable whether set theory can indeed have the role that is ascribed to it: it is far from clear that the axiom of choice is correctly regarded as a set-theoretic principle at all, and similar doubts may be raised about other purported applications of set-theoretic principles in mathematics.