# Set Theory/Zermelo‒Fraenkel set theory

Definition (Zermelo‒Fraenkel set theory):

Zermelo‒Fraenkel set theory (ZF set theory) is the theory given by the following axioms (where most are written in the language of set theory ${\displaystyle {\mathcal {L}}}$):

1. Axiom of extensionality: ${\displaystyle \forall x,y:(x=y)\Leftrightarrow \forall z:(z\in x\Leftrightarrow z\in y)}$
2. Empty set: ${\displaystyle \exists x:\forall y:(y\notin x)}$
3. Pairing: ${\displaystyle \forall x:\forall y:\exists m:\forall u:(u\in m)\Leftrightarrow ((u=x)\vee (u=y))}$
4. Union: ${\displaystyle \forall x:\exists u:\forall y:(y\in u)\Leftrightarrow (\exists z:(y\in z)\wedge (z\in x))}$
5. Restricted comprehension schema: Given a set ${\displaystyle y}$ and a formula in the language of set theory ${\displaystyle \phi (x)}$. Then we have a set of all ${\displaystyle z\in y}$ such that ${\displaystyle \phi (z)}$ is true. We write this as the set ${\displaystyle \{z\in y\,|\,\phi (z)\}}$.
6. Replacement schema: Given a set ${\displaystyle z}$ and a formula in the language of set theory ${\displaystyle \phi (x,y)}$, such that for all ${\displaystyle x}$ there exists a unique ${\displaystyle y}$ such that ${\displaystyle \phi (x,y)}$ is true. Then there exists a set ${\displaystyle a}$ whose elements are all those ${\displaystyle y}$. We write this as the set ${\displaystyle \{y\,|\,\exists x\in z:\phi (x,y)\}}$.
7. Power set: ${\displaystyle \forall x:\exists y:\forall z:(z\in y)\Leftrightarrow (\forall a:(a\in z)\Rightarrow (a\in x))}$
8. Infinity: There exists an inductive set. An inductive set is one that contains an empty set, and if it contains ${\displaystyle x}$, then it contains ${\displaystyle x\cup \{x\}}$.

Note that in the Infinity axiom, we said that the inductive set contains an' empty set. We will soon see that there is in fact a unique empty set, so we can change our definition of an inductive set to one that contains the' empty set.