Definition:Zermelo-Fraenkel Axioms

Axioms
The Zermelo-Fraenkel axioms are the most well-known basis for axiomatic set theory.

There is no standard numbering for them, and their exact formulation varies. Certain axioms can in fact be derived from other axioms, so their status as "axioms" can be questioned.

The axioms are as follows:

The Axiom of Foundation
The above axioms taken together as a system, but without the Axiom of Choice below, is called ZF set theory. The validity of the AC is still debated.

The Axiom of Choice
The system of the axioms of ZF set theory in combination with the Axiom of Choice is known as ZFC set theory; ZF plus Choice.