axiom schema of separation

Let $\varphi (u,p)$ be a formula. For any $X$ and $p$, there exists a set $Y=\{u\in X:\varphi (u,p)\}$.

The Axiom Schema of Separation is an axiom schema of Zermelo-Fraenkel set theory. Note that it represents infinitely many individual axioms, one for each formula $\varphi$. In symbols, it reads:

$$\forall X\forall p\exists Y\forall u(u\in Y\leftrightarrow u\in X\wedge \varphi (u,p)).$$

By Extensionality, the set $Y$ is unique.

The Axiom Schema of Separation implies that $\varphi$ may depend on more than one parameter $p$.

We may show by induction that if $\varphi (u,p_1,\mathrm{\dots },p_n)$ is a formula, then

$$\forall X\forall p_1\mathrm{\cdots }\forall p_n\exists Y\forall u(u\in Y\leftrightarrow u\in X\wedge \varphi (u,p_1,\mathrm{\dots },p_n))$$

holds, using the Axiom Schema of Separation and the Axiom of Pairing.

Another consequence of the Axiom Schema of Separation is that a subclass of any set is a set. To see this, let $𝐂$ be the class $𝐂=\{u:\varphi (u,p_1,\mathrm{\dots },p_n)\}$. Then

$$\forall X\exists Y(𝐂\cap X=Y)$$

holds, which means that the intersection of $𝐂$ with any set is a set. Therefore, in particular, the intersection of two sets $X\cap Y=\{x\in X:x\in Y\}$ is a set. Furthermore the difference of two sets $X-Y=\{x\in X:x\notin Y\}$ is a set and, provided there exists at least one set, which is guaranteed by the Axiom of Infinity, the empty set is a set. For if $X$ is a set, then $\mathrm{\varnothing }=\{x\in X:x\ne x\}$ is a set.

Moreover, if $𝐂$ is a nonempty class, then $\bigcap 𝐂$ is a set, by Separation. $\bigcap 𝐂$ is a subset of every $X\in 𝐂$.

Lastly, we may use Separation to show that the class of all sets, $V$, is not a set, i.e., $V$ is a proper class. For example, suppose $V$ is a set. Then by Separation

$$V^{\prime }=\{x\in V:x\notin x\}$$

is a set and we have reached a Russell paradox.