Model Theory
In 1954 Alfred Tarski proposed the name theory of models for the study of "mutual relations between sentences of formalized theories and mathematical [structures] in which these sentences hold." This definition hides a program that was to apply metamathematical results (particularly the Compactness Theorem of first-order logic) in what Abraham Robinson in 1950 had called "the development of actual mathematics." Anatolii I. Mal'tsev had launched this program in the Soviet Union in 1940, but communications were bad in this period and the program started afresh in the late 1940s with Tarski in the United States and Robinson in Britain. Mathematical model theory in the sense of this program has been remarkably successful, particularly in its applications to group theory and geometry, and it has far outgrown Tarski's initial definition of the theory of models.
Tarski's definition rested on the fact that one can use formal languages to define classes of structures. For mathematical applications it has turned out to be just as important that one can use formal languages to define sets and relations within a single structure. But at its base, model theory is more general even than this. Arguably it stands in the same relation to the traditional theory of definitions as modern proof theory stands to the traditional theory of syllogisms.
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