The ZF axioms

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.

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