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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| axiom schema of separation canonical | 1 |
| axiom schema of specification | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T694042 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: axiom schema of separation Context triple: [Zermelo–Fraenkel set theory, hasAxiom, axiom schema of separation]
-
A.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
-
B.
von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
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C.
von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
-
D.
Russell’s paradox
Russell’s paradox is a foundational logical contradiction in naive set theory that reveals problems with sets that contain themselves, leading to major developments in modern logic and the axiomatization of set theory.
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E.
von Neumann universe
The von Neumann universe is a cumulative, well-founded hierarchy of sets used as a standard model of the set-theoretic universe in axiomatic set theory.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: axiom schema of separation Target entity description: 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.
-
A.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
-
B.
von Neumann–Bernays–Gödel set theory
Von Neumann–Bernays–Gödel set theory is an axiomatic set theory extending Zermelo–Fraenkel set theory by formally distinguishing between sets and classes, widely used in foundational studies of mathematics.
-
C.
von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
-
D.
Russell’s paradox
Russell’s paradox is a foundational logical contradiction in naive set theory that reveals problems with sets that contain themselves, leading to major developments in modern logic and the axiomatization of set theory.
-
E.
von Neumann universe
The von Neumann universe is a cumulative, well-founded hierarchy of sets used as a standard model of the set-theoretic universe in axiomatic set theory.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: axiom schema of separation Description of subject: 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.
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