Ernst Zermelo and Abraham Fraenkel proposed the following axioms as a foundation for what is now called Zermelo-Fraenkel set theory, or ZF. If this set of axioms are accepted along with the Axiom of Choice, it is often denoted ZFC.

• Equality of sets: If $X$ and $Y$ are sets, and $x \in X$ iff $x \in Y$ then $X = Y$
• Pair set: If $X$ and $Y$ are sets, then there is a set $Z$ containing only $X$ and $Y$
• Union over a set: If $X$ is a set, then there exists a set that contains every element of each $x \in X$
• Axiom of power set: If $X$ is a set, then there exists a set $\mathcal{P}(x)$ with the property that $Y \in \mathcal{P}(x)$ iff any element $y \in Y$ is also in $X$
• Replacement axiom: Let $F(x,y)$ be some formula. If, for all $x$ there is exactly one $y$ such that $F(x,y)$ is true, then for any set $A$ there exists a set $B$ with the property that $b \in B$ iff there exists some $a \in A$ such that $F(a,b)$ is true.
• Regularity axiom: Let $F(x)$ be some formula. If there is some $x$ that makes $F(x)$ true, then there is a set $Y$ such that $F(Y)$ is true, but for no $y \in Y$ is $F(y)$ true.
• Existence of an infinite set: There exists a non-empty set $X$ with the property that, for any $x \in X$ there is some $y \in X$ such that $x \subseteq y$ but $x \neq y$
• Separation axiom: If $X$ is a set and $P$ is a condition on sets, there exists a set $Y$ whose members are precisely the members of $X$ satisfying $P$ (This axiom is also occasionally referred to as the subset axiom).