Zermelo–Fraenkel set theory
The Zermelo-Fraenkel axioms of set theory are the standard axioms of axiomatic set theory on which all of ordinary mathematics is based, except that they do not include the axiom of choice. The theory including the axiom of choice is often denoted by ZFC, as opposed to ZF -- here "Z" stands for "Zermelo", "F" for "Fraenkel", and "C" for "choice".
The system has infinite number of axioms because axiom schemata are used. An equivalent finite alternative system is given by the von Neumann-Bernays-Gödel axioms (NBG), which distinguish between classes and sets.
- Axiom of extension: Two sets are the same if and only if they have the same elements.
- Axiom of empty set: There is a set with no elements. We will use {} to denote this empty set.
- Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.
- Axiom of union: 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 infinity: There exists a set x such that {} is in x and whenever y is in x, so is the union y U {y}.
- Axiom of replacement: Given any set and any mapping, formally defined as a proposition P(x,y) where P(x,y) and P(x,z) implies y = z, there is a set containing precisely the images of the original set's elements.
- 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.
- Axiom of choice: Any product of nonempty sets is nonempty.