Ulam problem in set theory
E85414
The Ulam problem in set theory is a well-known question posed by Stanislaw Ulam concerning the structure and properties of measurable sets and functions, particularly in relation to homomorphisms and measure-theoretic regularity.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Ulam problem in set theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T718426 — 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: Ulam problem in set theory Context triple: [Stanislaw Ulam, notableWork, Ulam problem in set theory]
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A.
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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B.
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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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.
Surreal numbers
Surreal numbers are a class of numbers introduced by John H. Conway that form an extensive ordered field encompassing the real numbers, infinite quantities, and infinitesimals within a unified framework.
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E.
Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
- 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: Ulam problem in set theory Target entity description: The Ulam problem in set theory is a well-known question posed by Stanislaw Ulam concerning the structure and properties of measurable sets and functions, particularly in relation to homomorphisms and measure-theoretic regularity.
-
A.
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.
-
B.
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.
-
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.
Surreal numbers
Surreal numbers are a class of numbers introduced by John H. Conway that form an extensive ordered field encompassing the real numbers, infinite quantities, and infinitesimals within a unified framework.
-
E.
Remarks on the Foundations of Mathematics
Remarks on the Foundations of Mathematics is a posthumously published collection of Ludwig Wittgenstein’s later writings that critically examines the nature of mathematical truth, proof, and practice from a philosophical and language-centered perspective.
- F. None of above. chosen
Statements (31)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical problem
ⓘ
problem in measure theory ⓘ problem in set theory ⓘ |
| category |
open problems in mathematics
ⓘ
problems in set-theoretic measure theory ⓘ |
| concerns |
conditions for measure-theoretic regularity
ⓘ
existence of homomorphisms preserving measure-theoretic structure ⓘ structure of measurable functions ⓘ structure of measurable sets ⓘ |
| field |
functional analysis
ⓘ
measure theory ⓘ set theory ⓘ |
| hasAspect |
algebraic structure of measurable sets
ⓘ
measure-preserving homomorphisms ⓘ regularity of measures on set-theoretic structures ⓘ |
| involves |
measurable homomorphisms
ⓘ
set-theoretic properties of measures ⓘ σ-algebras of measurable sets ⓘ |
| mainSubject |
homomorphisms of measurable structures
ⓘ
measurable functions ⓘ measurable sets ⓘ measure-theoretic regularity ⓘ |
| motivation |
clarifying regularity assumptions in measure theory
ⓘ
understanding homomorphisms between measurable structures ⓘ |
| namedAfter |
Stanislaw Ulam
ⓘ
surface form:
Stanisław Ulam
|
| posedBy |
Stanislaw Ulam
ⓘ
surface form:
Stanisław Ulam
|
| relatedTo |
Stanislaw Ulam
ⓘ
surface form:
Stanisław Ulam
Ulam measurable cardinal ⓘ Ulam stability ⓘ regularity properties of measures ⓘ set-theoretic measure theory ⓘ |
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: Ulam problem in set theory Description of subject: The Ulam problem in set theory is a well-known question posed by Stanislaw Ulam concerning the structure and properties of measurable sets and functions, particularly in relation to homomorphisms and measure-theoretic regularity.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.