Editing Axiom schema of specification (section)

== In NBG class theory ==
{{Unreferenced section|date=June 2024}}
In [[von Neumann–Bernays–Gödel set theory]], a distinction is made between sets and [[class (set theory)|classes]]. A class {{mvar|C}} is a set if and only if it belongs to some class {{mvar|E}}. In this theory, there is a [[theorem]] schema that reads
<math display="block">\exists D \forall C \, ( [ C \in D ] \iff [ P (C) \land \exists E \, ( C \in E ) ] ) \,,</math>

that is,
{{block indent|There is a class {{mvar|D}} such that any class {{mvar|C}} is a member of {{mvar|D}} if and only if {{mvar|C}} is a set that satisfies {{mvar|P}}.}}

provided that the quantifiers in the predicate {{mvar|P}} are restricted to sets.

This theorem schema is itself a restricted form of comprehension, which avoids Russell's paradox because of the requirement that {{mvar|C}} be a set. Then specification for sets themselves can be written as a single axiom
<math display="block">\forall D \forall A \, ( \exists E \, [ A \in E ] \implies \exists B \, [ \exists E \, ( B \in E ) \land \forall C \, ( C \in B \iff [ C \in A \land C \in D ] ) ] ) \,,</math>

that is,
{{block indent|Given any class {{mvar|D}} and any set {{mvar|A}}, there is a set {{mvar|B}} whose members are precisely those classes that are members of both {{mvar|A}} and {{mvar|D}}.}}

or even more simply
{{block indent|The [[intersection (set theory)|intersection]] of a class {{mvar|D}} and a set {{mvar|A}} is itself a set {{mvar|B}}.}}

In this axiom, the predicate {{mvar|P}} is replaced by the class {{mvar|D}}, which can be quantified over. Another simpler axiom which achieves the same effect is
<math display="block">\forall A \forall B \, ( [ \exists E \, ( A \in E ) \land \forall C \, ( C \in B \implies C \in A ) ] \implies \exists E \, [ B \in E ] ) \,,</math>

that is,
{{block indent|A subclass of a set is a set.}}