In NBG Class Theory
In von Neumann–Bernays–Gödel set theory, a distinction is made between sets and classes. A class C is a set if and only if it belongs to some class E. In this theory, there is a theorem schema that reads:
that is:
- There is a class D such that any class C is a member of D if and only if C is a set that satisfies P.
This theorem schema is itself a restricted form of comprehension, which avoids Russell's paradox because of the requirement that C be a set. Then specification for sets themselves can be written as a single axiom:
that is:
- Given any class D and any set A, there is a set B whose members are precisely those classes that are members of both A and D;
or even more simply:
- The intersection of a class D and a set A is itself a set B.
In this axiom, the predicate P is replaced by the class D, which can be quantified over.
Read more about this topic: Axiom Schema Of Specification
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