Editing Gödel's completeness theorem (section)

==Completeness in other logics==
The completeness theorem is a central property of [[first-order logic]] that does not hold for all logics. [[Second-order logic]], for example, does not have a completeness theorem for its standard semantics (though does have the completeness property for [[Henkin semantics]]), and the set of logically valid formulas in second-order logic is not recursively enumerable. The same is true of all higher-order logics. It is possible to produce sound deductive systems for higher-order logics, but no such system can be complete.

[[Lindström's theorem]] states that first-order logic is the strongest (subject to certain constraints) logic satisfying both compactness and completeness.

A completeness theorem can be proved for [[modal logic]] or [[intuitionistic logic]] with respect to [[Kripke semantics]].