set theory
E85409
Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
All labels observed (4)
| Label | Occurrences |
|---|---|
| set theory canonical | 6 |
| Cantorian set theory | 1 |
| Mengenlehre | 1 |
| set theory with urelements | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T718408 — 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: set theory Context triple: [Stanislaw Ulam, fieldOfWork, set theory]
-
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.
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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.
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.
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D.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
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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: set theory Target entity description: Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
-
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.
Cantor’s theorem
Cantor’s theorem is a fundamental result in set theory stating that the power set of any set has a strictly greater cardinality than the set itself, implying there is no largest infinity.
-
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 (57)
| Predicate | Object |
|---|---|
| instanceOf |
branch of mathematical logic
ⓘ
branch of mathematics ⓘ |
| centralQuestion |
comparisons of infinite cardinalities
ⓘ
nature of infinity ⓘ |
| developedBy | Georg Cantor ⓘ |
| fieldOfStudy |
foundations of mathematics
ⓘ
sets ⓘ |
| hasAxiomSystem |
Kripke–Platek set theory
ⓘ
Zermelo–Fraenkel set theory ⓘ Zermelo–Fraenkel set theory ⓘ
surface form:
Zermelo–Fraenkel set theory with Choice
naive set theory ⓘ von Neumann–Bernays–Gödel set theory ⓘ |
| hasFoundationIn | axiomatic systems ⓘ |
| hasSubfield |
combinatorial set theory
ⓘ
descriptive set theory ⓘ determinacy theory ⓘ inner model theory ⓘ set-theoretic topology ⓘ |
| historicalPeriod | late 19th century ⓘ |
| includesConcept |
Aleph numbers
ⓘ
Russell’s paradox ⓘ
surface form:
Russell's paradox
axiom of choice ⓘ
surface form:
Zorn's lemma
axiom of choice ⓘ cardinal arithmetic ⓘ constructible universe ⓘ continuum hypothesis ⓘ empty set ⓘ forcing ⓘ large cardinals ⓘ ordinal arithmetic ⓘ universal set ⓘ well-ordering theorem ⓘ |
| isFoundationFor |
abstract algebra
ⓘ
analysis ⓘ category theory ⓘ functional analysis ⓘ measure theory ⓘ most of modern mathematics ⓘ topology ⓘ |
| languageUsed | first-order logic ⓘ |
| studies |
cardinal numbers
ⓘ
cardinality ⓘ collections of objects ⓘ functions ⓘ infinite sets ⓘ intersections ⓘ membership relations ⓘ ordinal numbers ⓘ power sets ⓘ relations ⓘ subsets ⓘ unions ⓘ |
| usesConcept |
intersection symbol ∩
ⓘ
membership symbol ∈ ⓘ power set operator P(X) ⓘ subset symbol ⊆ ⓘ union symbol ∪ ⓘ |
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: set theory Description of subject: Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
Referenced by (9)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Cantorian set theory
this entity surface form:
Mengenlehre
this entity surface form:
set theory with urelements