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axiom schema of separation (Axiom)

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:

$\displaystyle \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

$\displaystyle \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

$\displaystyle \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

$\displaystyle V' = \{ x \in V : x \notin x \} $
is a set and we have reached a Russell paradox.



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Other names:  separation schema, separation
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Cross-references: Russell paradox, proper class, subset, empty set, axiom of infinity, difference, intersection, class, subclass, consequence, axiom of pairing, induction, parameter, implies, extensionality, axioms, represents, Zermelo-Fraenkel set theory, axiom schema, formula
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This is version 15 of axiom schema of separation, born on 2003-06-24, modified 2003-06-25.
Object id is 4393, canonical name is AxiomSchemaOfSeparation.
Accessed 4215 times total.

Classification:
AMS MSC: 03E30 (Mathematical logic and foundations :: Set theory :: Axiomatics of classical set theory and its fragments)
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