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 below, is called Zermelo-Fraenkel set theory.

This is often seen abbreviated ZF.

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

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