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What (who) is axiom schema - definition
Axiom schema
A FORMULA IN THE METALANGUAGE OF AN AXIOMATIC SYSTEM IN WHICH ONE OR MORE SCHEMATIC VARIABLES APPEAR
Axiom scheme; Axiom schemata; Axiom-scheme; Finite axiomatization
In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.
Axiom schema of specification
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
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 schema of replacement
![Axiom schema of replacement: the image <math>F[A]</math> of the domain set <math>A</math> under the definable class function <math>F</math> is itself a set, <math>B</math>.](https://commons.wikimedia.org/wiki/Special:FilePath/Axiom schema of replacement.svg?width=200 "Axiom schema of replacement: the image <math>F[A]</math> of the domain set <math>A</math> under the definable class function <math>F</math> is itself a set, <math>B</math>.")
![Axiom schema of collection: the image <math>f[A]</math> of the domain set <math>A</math> under the definable class function <math>f</math> falls inside a set <math>B</math>.](https://commons.wikimedia.org/wiki/Special:FilePath/Codomain2 A B.SVG?width=200 "Axiom schema of collection: the image <math>f[A]</math> of the domain set <math>A</math> under the definable class function <math>f</math> falls inside a set <math>B</math>.")
IN SET THEORY, THE AXIOM SCHEMA THAT THE IMAGE OF A SET UNDER A DEFINABLE CLASS FUNCTION IS ALSO A SET
Axiom of replacement; Boundedness axiom; Axiom of boundedness; Axiom of collection; Irreplaceability; Axiom schema of collection; Replacement axiom; Axiom of substitution; Replacement implies separation
In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.
