Home > Zermelo-Fraenkel set theory

The axioms are the result of work by Thoralf Skolem in 1922, based on earlier work by Adolf Fraenkel in the same year, which was based on the axiom system put forth by Ernst Zermelo in 1908 ( Zermelo set theory).

The axiom system is written in first-order logic; it has an infinite number of axioms because an axiom schema is used. An alternative, finite system is given by the von Neumann-Bernays-Gödel axioms (NBG), which add the concept of a classIn set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets, for instance the classes in addition to that of a setThis article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now; it is "equivalent" in the sense that any theoremA theorem is a statement which can be proven true within some logical framework. Proving theorems is a central activity of mathematics. Note that 'theorem' is distinct from theory'. A theorem generally has a set-up a number of conditions, which may be lis about sets which can be proved in one system can be proven in the other.

The axioms of ZFC are:

• Axiom of extensionalitySet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality or axiom of extension is one of the axioms of Zermelo-Fraenkel set theory. In the formal language of the Zermelo-Fraen: Two sets are the same if and only if they have the same elements.
• Axiom of empty setSet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of empty set is one of the axioms of Zermelo-Fraenkel set theory. In the formal language of the Zermelo-Frankel axioms, the axiom reads:: There is a set with no elements. We will use {} to denote this empty set.
• Axiom of pairingSet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo-Fraenkel set theory. In the formal language of the Zermelo-Frankel axioms, the axiom reads: :: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.
• Axiom of unionSet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of union is one of the axioms of Zermelo-Fraenkel set theory, stating that, for any two sets, there is a set that contains exactly the e: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
• Axiom of infinitySet theory In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of infinity is one of the axioms of Zermelo-Fraenkel set theory. In the formal language of the Zermelo-Fraenkel axioms, the axiom reads:: There exists a set x such that {} is in x and whenever y is in x, so is the union y ∪ {y}.
• Axiom of replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y₁) and P(x,y₂) implies y₁ = y₂, there is a set containing precisely the images of the original set's elements. (This is an axiom schema.)
• Axiom of power set: Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x.
• Axiom of regularity: Every non-empty set x contains some element y such that x and y are disjoint sets.
• Axiom of separation (or subset axiom): Given any set and any proposition P(x), there is a subset of the original set containing precisely those elements x for which P(x) holds. (This is an axiom schema.)
• Axiom of choice: Given any set of mutually exclusive non-empty sets, there exists at least one set that contains exactly one element in common with each of the non-empty sets.

While most metamathematicians believe that these axioms are consistent (in the sense that no contradiction can be derived from them), this has not been proved. In fact, since they are the basis of ordinary mathematics, their consistency (if true) cannot be proved in ordinary mathematics; this is a consequence of Gödel's second incompleteness theorem. On the other hand, the consistency of ZFC can be proved by assuming the existence of an inaccessible cardinal.