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Axiom schema of specification - In NBG class theory | | Axiom schema of specification - In NBG class theory: Encyclopedia II - Axiom schema of specification - 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 iff 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 s ...
See also:Axiom schema of specification, Axiom schema of specification - Relation to the axiom schema of replacement, Axiom schema of specification - Unrestricted comprehension, Axiom schema of specification - In NBG class theory, Axiom schema of specification - In second order logic, Axiom schema of specification - In Quine's New Foundations | | | Axiom schema of specification, Axiom schema of specification - In NBG class theory, Axiom schema of specification - In Quine's New Foundations, Axiom schema of specification - In second order logic, Axiom schema of specification - Relation to the axiom schema of replacement, Axiom schema of specification - Unrestricted comprehension | | |
| | | Axiom schema of specification: Encyclopedia II - Axiom schema of specification - In NBG class theory
Axiom schema of specification - 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 iff 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.
Other related archivesAlternative Set Theory, Axioms of set theory, Given any, KPU, New Foundations, Russell's paradox, W.V.O. Quine, ZFC, Zermelo-Fraenkel set theory, alternative set theory, and, axiom of comprehension, axiom of empty set, axiom of extensionality, axiom of regularity, axiom schema, axiom schema of replacement, axiomatic set theory, axioms, classes, classical logic, computer science, empty set, formal language, functional predicate, if and only if, intersection, logic, mathematics, naive set theory, positive set theory, predicate, schema, second-order logic, semisets, set, set-builder notation, stratification, subclass, subset, theorem, there is, variable, von Neumann-Bernays-Gödel set theory
 Adapted from the Wikipedia article "In NBG class theory", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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