A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. History The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite. Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre: Bertrand Russell called a set a class: Naive set theory The foremost property of a set is that it can have elements, also called members. Two sets are equal when they have the same elements. More precisely, sets A and B are equal if every element of A is an element of B, and every element of B is an element of A; this property is called the extensionality of sets. The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed: Russell's paradox shows that the "set of all sets that do not contain themselves", i.e., , cannot exist. Cantor's paradox shows that "the set of all sets" cannot exist. Naïve set theory defines a set as any well-defined collection of distinct elements, but problems arise from the vagueness of the term well-defined. Axiomatic set theory In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by axioms. Axiomatic set theory takes the concept of a set as a primitive notion. The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using first-order logic. According to