Compactness and the Löwenheim–Skolem theorem
Two central theorems in model theory: Compactness guarantees a model exists if every finite subset is satisfiable; Löwenheim-Skolem relates model size to language cardinality.
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The statement of the theorem
Compactness: A set of sentences has a model iff every finite subset does. Löwenheim-Skolem: If a countable theory has an infinite model, it has models of every infinite cardinality.
Source: Wikipedia