Zermelo-Fraenkel axioms

The Zermelo-Fraenkel axioms form one of several possible formulations of axiomatic set theory.

The axioms

There are eight Zermelo-Fraenkel (ZF) axioms:

1. Axiom of extensionality: If X and Y have the same elements, then X=Y
2. Axiom of pairing: For any a and b there exists a set {a, b} that contains exactly a and b
3. Axiom schema of separation: If φ is a property with parameter p, then for any X and p there exists a set Y that contains all those elements u∈X that have the property φ; that is, the set Y={u∈X|φ(u, p)}
4. Axiom of union: For any set X there exists a set Y=∪X, the union of all elements of X
5. Axiom of power set: For any X there exists a set Y=P(X), the set of all subsets of X
6. Axiom of infinity: There exists an infinite set
7. Axiom schema of replacement: If f is a function, then for any X there exists a set Y, denoted F(X) such that F(X)={f(x)|x∈X}
8. Axiom of regularity: Every nonempty set has an ∈-minimal element

If to these is added the axiom of choice, the theory is designated as the ZFC theory:

 9. Axiom of choice: Every family of nonempty sets has a choice function

For further discussion of these axioms, see the bibliography.