Zermelo and Fraenkel expressed their axioms in the formal language of set theory, but we can translate them into ordinary language.

**Axiom 1:** Two sets are equal if and only if they have the same elements.

**Axiom 2:** There is a set with no elements called the *empty set*.

**Axiom 3:** For any sets $A$ and $B$ there is a set whose only elements are $A$ and $B$. (Note that taking $A=B$, this gives you the possibility of forming the singleton set $\{A\}$.)

**Axiom 4:** For any set $A$ there's a set whose elements are the elements of the elements of A. (If $A=\{B,C\}$ then this set is what's called the *union* $B \cup C$ of $B$ and $C$, whose elements are all the elements of $B$ together with the elements of $C$. For example if $B=\{1,2\}$ and $C=\{3,4\},$ then $B \cup C= \{1,2,3,4\}$.)

**Axiom 5:** For any set $A$ there is a set, called the *power set* of $A$, whose elements are the subsets of $A.$ (A subset of $A$ is a set whose elements are all contained in $A$. Power sets are an important ingredient of Cantor's hierarchy of infinities.)

**Axiom 6:** For any function $f$ defined on a set $A$, the values $f(a),$ where $a$ is an element of $A,$ also form a set.

**Axiom 7:** Every non-empty set $A$ has a member whose elements are all different from the elements of $A$. (This axiom insures that
if you take an element $B$ of $A$ and then an element $C$ of $B$, and so on, this process stops after finitely many steps.)

**Axiom 8:** There is an infinite set such that if $A$ is in this set, the set $A\cup \{A\}$ is also in the set.