Grothendieck universe
E98601
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Grothendieck universe canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T738058 — 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: Grothendieck universe Context triple: [von Neumann universe, relatedConcept, Grothendieck universe]
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A.
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.
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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.
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.
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D.
Morse–Kelley set theory by class–set distinction
Morse–Kelley set theory by class–set distinction is a foundational system that avoids certain set-theoretic paradoxes by rigorously distinguishing between sets and proper classes within a powerful axiomatic framework.
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E.
Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
- 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: Grothendieck universe Target entity description: A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
-
A.
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.
-
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.
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.
-
D.
Morse–Kelley set theory by class–set distinction
Morse–Kelley set theory by class–set distinction is a foundational system that avoids certain set-theoretic paradoxes by rigorously distinguishing between sets and proper classes within a powerful axiomatic framework.
-
E.
Zermelo set theory
Zermelo set theory is an early axiomatic system for set theory, introduced by Ernst Zermelo to rigorously formalize the concept of sets and avoid known paradoxes.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
ⓘ
set-theoretic construct ⓘ tool in category theory ⓘ |
| advantage | allows working inside a single set that behaves like a miniature universe of sets ⓘ |
| alternativeTo |
Morse–Kelley set theory by class–set distinction
ⓘ
surface form:
Morse–Kelley set theory
von Neumann–Bernays–Gödel set theory ⓘ |
| assumption | existence is not provable in ZFC without large cardinal axioms ⓘ |
| contains |
all usual small mathematical objects used in a given context
ⓘ
empty set ⓘ finite sets of its elements ⓘ natural numbers ⓘ power sets of its elements ⓘ unions of families of its elements indexed by its elements ⓘ |
| correspondsTo | strongly inaccessible cardinal ⓘ |
| definition |
a set U such that if (x_i)_{i∈I} is a family of elements of U indexed by I ∈ U then ⋃_{i∈I} x_i ∈ U
ⓘ
a set U such that if x ∈ U and y ∈ x then y ∈ U ⓘ a set U such that if x ∈ U then P(x) ∈ U ⓘ a set U such that if x,y ∈ U then {x,y} ∈ U ⓘ |
| equivalentCondition | U is a Grothendieck universe iff U = V_κ for some strongly inaccessible κ (in ZFC + inaccessibles) ⓘ |
| field |
category theory
ⓘ
set theory ⓘ |
| formalization | often added to ZFC as an extra axiom scheme ⓘ |
| hasModel | V_κ for an inaccessible cardinal κ ⓘ |
| implies | if x ∈ U then the transitive closure of x is contained in U ⓘ |
| limitation | existence of nontrivial universes is independent of ZFC ⓘ |
| namedAfter | Alexander Grothendieck ⓘ |
| philosophicalRole | provides a relative notion of smallness and largeness in mathematics ⓘ |
| property |
closed under pairing
ⓘ
closed under power set ⓘ closed under unions of families indexed by elements of the universe ⓘ transitive set ⓘ |
| relatedConcept |
cumulative hierarchy
ⓘ
inaccessible cardinal ⓘ large cardinal ⓘ universe axiom ⓘ |
| requires | inaccessible cardinal axiom for nontrivial examples ⓘ |
| role | separates size issues from structural arguments in category theory ⓘ |
| usedFor |
avoiding set-theoretic paradoxes
ⓘ
formalizing universes of discourse in mathematics ⓘ foundations of category theory ⓘ handling large categories ⓘ |
| usedIn |
algebraic geometry
ⓘ
higher category theory ⓘ homological algebra ⓘ topos theory ⓘ Éléments de géométrie algébrique ⓘ |
| usedToDefine |
locally small category
ⓘ
small category ⓘ universe of sets for a category ⓘ |
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: Grothendieck universe Description of subject: A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.