information:classification, linguistic:paraphrasing, linguistic:voice_change, information:extraction
History

Model theory as a subject has existed since approximately the middle of the 20th century, and the name was coined by Alfred Tarski, a member of the Lwów–Warsaw school, in 1954. However some earlier research, especially in mathematical logic, is often regarded as being of a model-theoretical nature in retrospect. The first significant result in what is now model theory was a special case of the downward Löwenheim–Skolem theorem, published by Leopold Löwenheim in 1915. The compactness theorem was implicit in work by Thoralf Skolem, but it was first published in 1930, as a lemma in Kurt Gödel 's proof of his completeness theorem. The Löwenheim–Skolem theorem and the compactness theorem received their respective general forms in 1936 and 1941 from Anatoly Maltsev.The development of model theory as an independent discipline was brought on by Alfred Tarski during the interbellum. Tarski's work included logical consequence, deductive systems, the algebra of logic, the theory of definability, and the semantic definition of truth, among other topics. His semantic methods culminated in the model theory he and a number of his Berkeley students developed in the 1950s and '60s.

In the further history of the discipline, different strands began to emerge, and the focus of the subject shifted. In the 1960s, techniques around ultraproducts became a popular tool in model theory. At the same time, researchers such as James Ax were investigating the first-order model theory of various algebraic classes, and others such as H. Jerome Keisler were extending the concepts and results of first-order model theory to other logical systems. Then, inspired by Morley's problem, Shelah developed stability theory. His work around stability changed the complexion of model theory, giving rise to a whole new class of concepts. This is known as the paradigm shift. Over the next decades, it became clear that the resulting stability hierarchy is closely connected to the geometry of sets that are definable in those models; this gave rise to the subdiscipline now known as geometric stability theory. An example of an influential proof from geometric model theory is Hrushovski 's proof of the Mordell–Lang conjecture for function fields.