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- Compactness_theorem abstract "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. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent. The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces; hence, the theorem's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection.The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them.".
- Compactness_theorem wikiPageID "152207".
- Compactness_theorem wikiPageRevisionID "589870034".
- Compactness_theorem hasPhotoCollection Compactness_theorem.
- Compactness_theorem subject Category:Metatheorems.
- Compactness_theorem subject Category:Model_theory.
- Compactness_theorem subject Category:Theorems_in_the_foundations_of_mathematics.
- Compactness_theorem type Abstraction100002137.
- Compactness_theorem type Communication100033020.
- Compactness_theorem type Message106598915.
- Compactness_theorem type Proposition106750804.
- Compactness_theorem type Statement106722453.
- Compactness_theorem type Theorem106752293.
- Compactness_theorem type TheoremsInTheFoundationsOfMathematics.
- Compactness_theorem comment "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. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.".
- Compactness_theorem label "Compactness theorem".
- Compactness_theorem label "Kompaktheitssatz".
- Compactness_theorem label "Teorema da compacidade".
- Compactness_theorem label "Teorema de compacidad".
- Compactness_theorem label "Teorema di compattezza (logica matematica)".
- Compactness_theorem label "Théorème de compacité".
- Compactness_theorem label "Twierdzenie o zwartości".
- Compactness_theorem label "紧致性定理".
- Compactness_theorem sameAs Věta_o_kompaktnosti.
- Compactness_theorem sameAs Kompaktheitssatz.
- Compactness_theorem sameAs Teorema_de_compacidad.
- Compactness_theorem sameAs Théorème_de_compacité.
- Compactness_theorem sameAs Teorema_di_compattezza_(logica_matematica).
- Compactness_theorem sameAs 콤팩트성_정리.
- Compactness_theorem sameAs Twierdzenie_o_zwartości.
- Compactness_theorem sameAs Teorema_da_compacidade.
- Compactness_theorem sameAs m.013tmj.
- Compactness_theorem sameAs Q1149458.
- Compactness_theorem sameAs Q1149458.
- Compactness_theorem sameAs Compactness_theorem.
- Compactness_theorem wasDerivedFrom Compactness_theorem?oldid=589870034.
- Compactness_theorem isPrimaryTopicOf Compactness_theorem.