A set of first-order sentences has a model if and only if every finite subset of it has a model. One of the two central theorems of first-order model theory alongside completeness, it allows the construction of nonstandard models, including nonstandard models of arithmetic and analysis.
Facts
StatementA set of first-order sentences has a model if and only if every finite subset of that set has a model. 1 Proof YearGodel proved the countable-language case in 1930; the same source's History section separately credits Anatoly Maltsev with the uncountable case in 1936. Classification
Statement FormCharacterization Theorem 1 Connections
Sources
1. Compactness theorem (Wikipedia)
Wikimedia FoundationLead section, first sentence
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model.
History section, first sentence
Kurt Gödel proved the countable compactness theorem in 1930.
Lead section, statement-form reference
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model.
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