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AXIOM SCHEMA
Axiom of specification; Axiom of separation; Axiom schema of separation; Axiom schema of comprehension; Axiom of comprehension; Unrestricted comprehension; Axiom of abstraction; Axiom of subsets; Axioms of subsets; Subset axiom; Axiom schema of restricted comprehension; Comprehension axiom; Aussonderungsaxiom; Axiom schema of unrestricted comprehension; Unrestricted comprehension principle
Axiom of Comprehension
<mathematics> An axiom schema of set theory which states:
if P(x) is a property then
x : P
is a set. I.e. all the things with some property form a set.
Acceptance of this axiom leads to Russell's Paradox which is
why Zermelo set theory replaces it with a restricted form.
(1995-03-31)
Axiom schema of specification
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.
Axiom of extensionality
AXIOM OF ZERMELO–FRAENKEL SET THEORY ASSERTING THAT SET EQUALITY IS DETERMINED BY THE MEMBERSHIP RELATION
Axiom of extension; Axiom of Extensionality; Axiom extensionality; Extensionality axiom; Axiom of equality
In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements are the same set.
ويكيبيديا
Axiom schema of specification
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.
Some mathematicians call it the axiom schema of comprehension, although others use that term for unrestricted comprehension, discussed below.
Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory.
