In mathematical logic, the Loewenheim-Skolem theorem is a theorem on the existence and cardinality of models, named after Leopold Loewenheim and Thoralf Skolem. It implies that if a countable first-order theory has an infinite model, then it has models of every infinite cardinality, so no first-order theory with an infinite model can be unique up to isomorphism, because first-order theories are unable to control the cardinality of their infinite models. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementThe Loewenheim-Skolem theorem states that if a countable first-order theory has an infinite model, then for every infinite cardinal number it also has a model of that size, so no first-order theory with an infinite model can have a unique model up to isomorphism. 2 Classification
Statement Form Connections
Sources
1. Wikipedia: Löwenheim-Skolem theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism.
2. Loewenheim-Skolem theorem (Wikipedia)
Wikimedia FoundationLöwenheim-Skolem theorem, lead section, sentence 2
It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism.
Löwenheim-Skolem theorem, Historical notes section
The first significant result in what later became model theory was Löwenheim's theorem in Leopold Löwenheim's publication "Über Möglichkeiten im Relativkalkül" [On possibilities in the calculus of relatives] (1915):
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