axiom schema of separation

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

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 $\phi$ . 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.

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

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.

Title	axiom schema of separation
Canonical name	AxiomSchemaOfSeparation
Date of creation	2013-03-22 13:42:46
Last modified on	2013-03-22 13:42:46
Owner	Sabean (2546)
Last modified by	Sabean (2546)
Numerical id	18
Author	Sabean (2546)
Entry type	Axiom
Classification	msc 03E30
Synonym	separation schema
Synonym	separation