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For standard first-order logic, every semantically valid consequence is derivable in a sound and complete proof calculus.
This is Gödel’s completeness theorem for first-order logic, not either of Gödel’s incompleteness theorems and not semantic completeness of a particular theory.
wp-godel-completeness-theoremnlab-completeness-theoremopenlogic-model-theoryEach relation below ends at this concept.
No direct relations are authored in this direction.
Each relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
If an unsatisfiable theory entailed a contradiction, completeness would give a finite formal proof using only finitely many premises; contraposition yields compactness.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-compactness-theoremnlab-compactness-theoremopenlogic-model-theory