axiom schema of separation
Let be a formula. For any and , there exists a set .
By Extensionality, the set is unique.
The Axiom Schema of Separation implies that may depend on more than one parameter .
We may show by induction that if is a formula, then
holds, using the Axiom Schema of Separation and the Axiom of Pairing.
holds, which means that the intersection of with any set is a set. Therefore, in particular, the intersection of two sets is a set. Furthermore the difference of two sets 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 is a set, then is a set.
Moreover, if is a nonempty class, then is a set, by Separation. is a subset of every .
Lastly, we may use Separation to show that the class of all sets, , is not a set, i.e., is a proper class. For example, suppose is a set. Then by Separation
is a set and we have reached a Russell paradox.
|Title||axiom schema of separation|
|Date of creation||2013-03-22 13:42:46|
|Last modified on||2013-03-22 13:42:46|
|Last modified by||Sabean (2546)|