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# axiom schema of separation

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 $\phi$. In symbols, it reads:

$\forall X\forall p\exists Y\forall u(u\in Y\leftrightarrow u\in X\land\phi(u,p% )).$ |

By Extensionality, the set $Y$ is unique.

We may show by induction that if $\phi(u,p_{1},\ldots,p_{n})$ is a formula, then

$\forall X\forall p_{1}\cdots\forall p_{n}\exists Y\forall u(u\in Y% \leftrightarrow u\in X\land\phi(u,p_{1},\ldots,p_{n}))$ |

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 $\mathbf{C}$ be the class $\mathbf{C}=\{u:\phi(u,p_{1},\ldots,p_{n})\}$. Then

$\forall X\exists Y(\mathbf{C}\cap X=Y)$ |

holds, which means that the intersection of $\mathbf{C}$ 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 $\emptyset=\{x\in X:x\neq x\}$ is a set.

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

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.

## Mathematics Subject Classification

03E30*no label found*

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