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Axiom schema of specification - Unrestricted comprehension | | Axiom schema of specification - Unrestricted comprehension: Encyclopedia II - Axiom schema of specification - Unrestricted comprehension | | The axiom schema of unrestricted comprehension reads:
that is:
There exists a set B whose members are precisely those objects that satisfy the predicate P.
This set B is again unique, and is usually denoted as {C : P(C)}.
This axiom schema was tacitly used in the early days of naive set theory, before a strict axiomatisation was adopted. Unfortunately, it leads directly to Russell's paradox by taking P(C) to be (< ...
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 - Unrestricted comprehension
Axiom schema of specification - Unrestricted comprehension
The axiom schema of unrestricted comprehension reads:
that is:
There exists a set B whose members are precisely those objects that satisfy the predicate P.
This set B is again unique, and is usually denoted as {C : P(C)}.
This axiom schema was tacitly used in the early days of naive set theory, before a strict axiomatisation was adopted. Unfortunately, it leads directly to Russell's paradox by taking P(C) to be (C is not in C). Therefore, no useful axiomatisation of set theory can use unrestricted comprehension, at least not with classical logic.
Accepting only the axiom schema of specification was the beginning of axiomatic set theory. Most of the other Zermelo-Fraenkel axioms (but not the axiom of extensionality or the axiom of regularity) then became necessary to serve as an additional replacement for the axiom schema of comprehension; each of these axioms states that a certain set exists, and defines that set by giving a predicate for its members to satisfy.
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 "Unrestricted comprehension", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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