Here we present the axioms that are part of [Zermelo-Fraenkel set theory axioms](Axioms%20of%20set%20theory.md) that should be taken to be the formal definition of a [set.](Sets.md) These axioms are compatible with the [Primitive notion](Primitive%20notion.md)s that form our intuition of what a set is. That is, that a set is [a collection of things.](Sets%20(index).md#Sets) These axioms are: * _[The Axiom of extension](Axioms%20of%20set%20theory.md#Axiom%20of%20extension)_  * _[The Axiom of separation](Axioms%20of%20set%20theory.md#Axiom%20of%20separation)_  * _[Axiom of union](Axioms%20of%20set%20theory.md#Axiom%20of%20union)_ The first three axioms serve to prove the [existence of the null set.](The%20null%20set.md#Proof%20of%20the%20existence%20of%20the%20null%20set) Below are additional axioms needed to define the notion of a set containing elements. #MathematicalFoundations/SetTheory