axiom schema of separation (original) (raw)

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

We may show by induction that if ϕ⁢(u,p1,…,pn) is a formula, then

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:ϕ⁢(u,p1,…,pn)}. Then

holds, which means that the intersection of 𝐂 with any set is a set. Therefore, in particular, the intersection of two sets X∩Y={x∈X:x∈Y} is a set. Furthermore the difference of two sets X-Y={x∈X:x∉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 ∅={x∈X:x≠x} is a set.

Moreover, if 𝐂 is a nonempty class, then ⋂𝐂 is a set, by Separation. ⋂𝐂 is a subset of every X∈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

is a set and we have reached a Russell paradox.