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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Observed surface forms (7)

Surface form	Occurrences
Löwenheim–Skolem theorem	2
Skolem paradox	0
downward Löwenheim–Skolem theorem	1
Skolem's paradox	2
upward Löwenheim–Skolem theorem	0
Skolem hull	1
Skolem paradox about countable models of set theory	1

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 ⓘ

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