Definition:Zermelo-Fraenkel Axioms

Definition
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 of these 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 Zermelo-Fraenkel set theory.

This is often seen abbreviated ZF.

The Axiom of Choice
Whether or not the axiom of choice (AC) is accepted is more or less a philosophical position.

The system of ZF set theory in combination with the axiom of choice is known as ZFC set theory: ZF plus Choice.