Löwenheim–Skolem theorem (via additional arguments)

E446857

apparent paradox in set theory result in model theory theorem in mathematical logic theorem in model theory

The Löwenheim–Skolem theorem is a fundamental result in model theory stating that any first-order theory with an infinite model has models of all infinite cardinalities, leading to the so-called Skolem paradox about the existence of countable models of set theory.

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All labels observed (6)

Label	Occurrences
Löwenheim–Skolem theorem	2
Skolem's paradox	2
Löwenheim–Skolem theorem (via additional arguments) canonical	1
Skolem hull	1
Skolem paradox about countable models of set theory	1
downward Löwenheim–Skolem theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T4492883 — 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: Löwenheim–Skolem theorem (via additional arguments)
Context triple: [completeness theorem for first-order logic, implies, Löwenheim–Skolem theorem (via additional arguments)]

A. Tarski–Mostowski–Robinson theorem
The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
B. Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
C. Hamilton’s compactness theorem
Hamilton’s compactness theorem is a fundamental result in geometric analysis that provides conditions under which a sequence of Riemannian manifolds with controlled curvature and injectivity radius admits a smoothly convergent subsequence.
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. completeness theorem for first-order logic
The completeness theorem for first-order logic is a fundamental result in mathematical logic, proved by Kurt Gödel, which states that every logically valid first-order formula is provable from the axioms of first-order logic.
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: Löwenheim–Skolem theorem (via additional arguments)
Target entity description: The Löwenheim–Skolem theorem is a fundamental result in model theory stating that any first-order theory with an infinite model has models of all infinite cardinalities, leading to the so-called Skolem paradox about the existence of countable models of set theory.

A. Tarski–Mostowski–Robinson theorem
The Tarski–Mostowski–Robinson theorem is a fundamental result in model theory that characterizes when a class of structures is first-order axiomatizable, linking definability properties with closure under ultraproducts and isomorphisms.
B. Tarski's undefinability theorem
Tarski's undefinability theorem is a fundamental result in mathematical logic showing that, in sufficiently strong formal systems, the notion of truth for the language of the system cannot be defined within that same language.
C. Hamilton’s compactness theorem
Hamilton’s compactness theorem is a fundamental result in geometric analysis that provides conditions under which a sequence of Riemannian manifolds with controlled curvature and injectivity radius admits a smoothly convergent subsequence.
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. completeness theorem for first-order logic
The completeness theorem for first-order logic is a fundamental result in mathematical logic, proved by Kurt Gödel, which states that every logically valid first-order formula is provable from the axioms of first-order logic.
F. None of above. chosen

Statements (47)

Predicate	Object
instanceOf	apparent paradox in set theory ⓘ result in model theory ⓘ theorem in mathematical logic ⓘ theorem in model theory ⓘ theorem in model theory ⓘ
appliesTo	any first-order theory with an infinite model ⓘ first-order Peano arithmetic ⓘ first-order Zermelo–Fraenkel set theory NERFINISHED ⓘ first-order theories of fields ⓘ
assumes	standard semantics for first-order logic ⓘ
cardinalityCondition	for uncountable languages, yields models of cardinalities bounded in terms of the language size ⓘ requires the language to be at most countable for the classical downward version ⓘ
concerns	existence of countable models of set theory that talk about uncountable sets ⓘ first-order logic ⓘ first-order theories ⓘ models of theories ⓘ
doesNotApplyTo	second-order logic with full semantics ⓘ
field	mathematical logic ⓘ model theory ⓘ
formalizes	existence of elementary submodels of smaller cardinality under certain conditions ⓘ
hasConsequence	no first-order theory with an infinite model can control the cardinality of all its models ⓘ no infinite structure is categorical in all infinite cardinalities in first-order logic ⓘ
hasPart	downward Löwenheim–Skolem theorem NERFINISHED ⓘ upward Löwenheim–Skolem theorem NERFINISHED ⓘ
historicalDevelopment	early form proved by Leopold Löwenheim in 1915 ⓘ refined and simplified by Thoralf Skolem in the 1920s ⓘ
implies	existence of countable models for theories with uncountable models ⓘ existence of models of all infinite cardinalities for certain theories ⓘ
influenced	development of axiomatic set theory ⓘ philosophy of mathematics discussions about relativity of set-theoretic notions ⓘ
involves	Skolem functions NERFINISHED ⓘ Skolem hulls NERFINISHED ⓘ elementary substructures ⓘ
namedAfter	Leopold Löwenheim NERFINISHED ⓘ Thoralf Skolem NERFINISHED ⓘ
relatedTo	Löwenheim–Skolem theorem NERFINISHED ⓘ Skolem paradox ⓘ compactness theorem NERFINISHED ⓘ completeness theorem for first-order logic ⓘ
requires	compactness of first-order logic for some proofs ⓘ
shows	cardinality of a model of a first-order theory is not uniquely determined by the theory if it has an infinite model ⓘ first-order set theory has countable models if it has any infinite model ⓘ
states	If a first-order theory has an infinite model then it has a countable model ⓘ If a first-order theory has an infinite model then it has models of arbitrarily large infinite cardinalities ⓘ
usedIn	classification of models by cardinality ⓘ model-theoretic analysis of set theory ⓘ proofs of non-categoricity of many first-order theories in infinite cardinals ⓘ

How these facts were elicited

Referenced by (8)

Full triples — surface form annotated when it differs from this entity's canonical label.

completeness theorem for first-order logic → implies → Löwenheim–Skolem theorem (via additional arguments) ⓘ

Thoralf Skolem → notableWork → Löwenheim–Skolem theorem (via additional arguments) ⓘ

this entity surface form: Skolem's paradox

Thoralf Skolem → notableWork → Löwenheim–Skolem theorem (via additional arguments) ⓘ

this entity surface form: Löwenheim–Skolem theorem

Thoralf Skolem → notableWork → Löwenheim–Skolem theorem (via additional arguments) ⓘ

this entity surface form: Skolem hull

Thoralf Skolem → knownFor → Löwenheim–Skolem theorem (via additional arguments) ⓘ

this entity surface form: Skolem's paradox

Thoralf Skolem → knownFor → Löwenheim–Skolem theorem (via additional arguments) ⓘ

this entity surface form: Löwenheim–Skolem theorem

Thoralf Skolem → notableIdea → Löwenheim–Skolem theorem (via additional arguments) ⓘ

this entity surface form: downward Löwenheim–Skolem theorem

Thoralf Skolem → notableIdea → Löwenheim–Skolem theorem (via additional arguments) ⓘ

this entity surface form: Skolem paradox about countable models of set theory

All labels observed (6)

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Statements (47)

How these facts were elicited Show

Referenced by (8)

How this entity was disambiguated

How these facts were elicited