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

These are the de facto standard axioms for contemporary mathematics Axiom of extensionality Axiom of empty set Axiom of pairing Axiom of union Axiom of infinity Axiom schema of replacement Axiom of power set Axiom of regularity Axiom of separation Axiom schema of specification See also Zermelo set theory. ...

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List of axioms, List of axioms - Zermelo-Frankel axioms, List of axioms - Axiom of choice, List of axioms - Equivalents of AC, List of axioms - Weaker than AC, List of axioms - Alternates incompatible with AC, List of axioms - Other axioms of mathematical logic, List of axioms - Geometry, List of axioms - Other axioms

Read more here: » List of axioms: Encyclopedia II - List of axioms - Zermelo-Frankel axioms

With the Zermelo-Frankel axioms above, this makes up the system ZFC in which most mathematics is potentially formalisable List of axioms - Equivalents of AC. Hausdorff maximality theorem Well-ordering principle Zorn's lemma List of axioms - Weaker than AC. Axiom of countable choice Axiom of dependent choice Boolean prime ideal theorem Axiom of uniformization List of axioms - Alternates incompatible with ACSee also:

List of axioms, List of axioms - Zermelo-Frankel axioms, List of axioms - Axiom of choice, List of axioms - Equivalents of AC, List of axioms - Weaker than AC, List of axioms - Alternates incompatible with AC, List of axioms - Other axioms of mathematical logic, List of axioms - Geometry, List of axioms - Other axioms

Read more here: » List of axioms: Encyclopedia II - List of axioms - Axiom of choice

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

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

Read more here: » Axiom schema of specification: Encyclopedia II - Axiom schema of specification - Unrestricted comprehension