In proof theory, a formal system is said to be syntactically complete if and only if every closed sentence in the system is such that either it or its negation is provable in the system. In model theory, a formal system is said to be semantically…

The resulting “proof theory” was concerned primarily (though not exclusively) with the different kinds of proof that can be accomplished within formal systems.

…which is closely related to proof theory, must often be distinguished from semantics, which is closely related to model theory. Roughly speaking, syntax—as conceived in the philosophy of mathematics—is a branch of number theory, and semantics is a branch of set theory, which deals with the nature and relations of…

…formal system is known as proof theory. It is one of the main areas of systematic logical theory.

…axiomatic theory T is called proof theory, or metamathematics. It is premised upon the formulation of T as a formal axiomatic theory—i.e., the theory of inference (as well as T) must be axiomatized. It is then possible to present T in a purely symbolic form—i.e., as a formal language based…