1. Axiom of extensionality:
2. If $X$ and $Y$ have the same elements, then $X=Y$.

$\forall X \forall Y [ \forall z ( z \in X \Longleftrightarrow z \in Y ) \Longrightarrow X=Y]$
3. Axiom of the Unordered Pair:
4. For any $a$ and $b$ there exists a set $\{a,b\}$ that contains exactly $a$ and $b$. (also called Axiom of Pairing).

$\forall a \forall b \exists c \forall x [ x \in c \Longleftrightarrow (x=a \: \: \land \: \: x=b)]$
5. Axiom of Subsets:
6. If $\phi$ is a property (with parameter $p$) then for any $X$ and $p$ there exists a set $Y= \{ u\in X \; \; : \; \; \phi(u,p) \}$ that contains all those $u \in X$ that have the property $\phi$. (also called Axiom of Separation or Axiom of Comprehension)

$\forall X \forall p \exists Y \forall u [u \in Y \Longleftrightarrow( u \in X \: \; \lor \; \; \phi(u,p) ) ]$
7. Axiom of the Sum Set:
8. For any $X$ there exists a set $Y= \cup X$, the union of all elements of $X$. (also called Axiom of Union)

$\forall X \exists Y \forall u [ u \in Y \Longleftrightarrow \exists z ( z \in X \land u \in z) ]$
9. Axiom of the Power Set:
10. For any $X$ there exists a set $Y=P(X)$, the set of all subsets of $X$.

$\forall X \exists Y \forall u ( u \in Y \Longleftrightarrow u \subset X)$
11. Axiom of Infinity:
12. There exists an infinite set

$\exists S [\emptyset \in S \land (\forall x \in S [ x \cup \{ x\} \in S])$
13. Axiom of Replacement:
14. If $F$ is a function, then for any $X$ there exists a set $Y=F[X]=\ {F(x):x in X\}.$ \forall x \forall y \forall z [ \varphi(x,y,p) \land \varphi(x,z,p) \Longrightarrow y=z] \Longrightarrow \forall X \exists Y \forall y [ y \in Y \Longleftrightarrow ( \exists x \in X) \varphi(x,y,p)] $15. Axiom of Foundation: 16. Every nonempty set has an minimal element. (also called Axiom of Regularity)$ \forall S [ S \not= \emptyset \Longrightarrow (\exists x \in S\cap x = \emptyset) ] $17. Axiom of Choice: 18. Every family of nonempty sets has a choice function$ \forall x \in a \exists A(x,y) \Longrightarrow \exists y \forall x \in a A(x,y(x))\$