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 (< ...

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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

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The axiom schema of separation can almost be derived from the axiom schema of replacement. First, recall this axiom schema: for any functional predicate F in one variable that doesn't use the symbols A, B, C or D. Given a suitable predicate P for the axiom of specification, define the mapping F by F(D) = D if P(D) is true and F(D) = E if P(D) is false, where E is any member of ASee 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

Read more here: » Axiom schema of specification: Encyclopedia II - Axiom schema of specification - Relation to the axiom schema of replacement

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

Read more here: » Axiom schema of specification: Encyclopedia II - Axiom schema of specification - In NBG class theory