axiom schema of separation
E84443
The axiom schema of separation is a principle in set theory that guarantees the existence of subsets defined by properties or predicates, helping to avoid paradoxes by restricting unrestricted set formation.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| axiom schema of specification | 1 |
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
axiom schema
ⓘ
set theory axiom ⓘ |
| alsoKnownAs |
axiom schema of restricted comprehension
ⓘ
axiom schema of separation ⓘ
surface form:
axiom schema of specification
axiom schema of subset formation ⓘ |
| assumes | that all elements of the subset come from a given set A ⓘ |
| centralTo | modern axiomatic set theory ⓘ |
| compatibleWith | standard cumulative hierarchy view of sets ⓘ |
| componentOf |
Zermelo–Fraenkel set theory
ⓘ
surface form:
Zermelo–Fraenkel axioms
|
| contrastsWith | axiom schema of unrestricted comprehension ⓘ |
| dependsOn | prior existence of a set from which a subset is separated ⓘ |
| doesNotAllow | formation of a set of all objects satisfying a property without a bounding set ⓘ |
| doesNotEntail | existence of a universal set ⓘ |
| ensures | existence of subsets defined by formulas ⓘ |
| entails | every definable subclass of a set is a set ⓘ |
| expressedAs | ∀A ∀p1 … ∀pn ∃B ∀x (x ∈ B ↔ x ∈ A ∧ φ(x, p1, …, pn)) ⓘ |
| expressibleIn | first-order logic with equality and membership ⓘ |
| field | set theory ⓘ |
| formalizes | subset formation by property ⓘ |
| foundationFor | safe use of predicates in set formation ⓘ |
| hasConsequence | definable subclasses of the universe may fail to be sets ⓘ |
| hasForm | for any formula φ(x, p1, …, pn) and any set A, there is a set {x in A : φ(x, p1, …, pn)} ⓘ |
| implies | existence of subsets of any given set satisfying a given property ⓘ |
| influencedBy | paradoxes of naive set theory ⓘ |
| introducedBy | Ernst Zermelo ⓘ |
| introducedInContextOf | foundations of set theory ⓘ |
| introducedToSolve | logical paradoxes in naive set theory ⓘ |
| isInfiniteSchema | true ⓘ |
| logicalForm | infinite family of axioms, one for each formula φ ⓘ |
| prevents | construction of Russell set {x : x ∉ x} as a set ⓘ |
| purpose | to avoid set-theoretic paradoxes such as Russell's paradox ⓘ |
| quantifiesOver | formulas of the language of set theory ⓘ |
| relatedTo |
axiom of power set
ⓘ
axiom of replacement ⓘ |
| requires | a defining formula with parameters ⓘ |
| restricts | unrestricted set comprehension ⓘ |
| role | to limit comprehension to subsets of existing sets ⓘ |
| schemaOver | all formulas φ in the first-order language of set theory ⓘ |
| usedIn |
ZF
ⓘ
surface form:
ZFC
Zermelo set theory ⓘ Zermelo–Fraenkel set theory ⓘ |
| usedToProve |
existence of intersections of sets
ⓘ
existence of many standard set-theoretic constructions ⓘ existence of relative complements ⓘ |
| weakerThan | axiom schema of replacement ⓘ |
| yearIntroducedApprox | 1908 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
axiom schema of specification