The Reasonable Effectiveness of Model Theory in Mathematics

In this article we first provide some background on why (Sections 1 and 2) the applications of model theory across mathematics are reasonable. Section 3 describe some of these applications. While we allude to a number of well-known results over the last seventy years, we focus on three areas that have developed in the last five. We survey the parallel developments of certain combinatorial notions in learning theory (Section 3.1) and in functional analysis (Section 3.2) with fundamental notions of stability theory. Section 4 applies the study of trivial weakly minimal sets, structures very near the base of the stability hierarchy, to count the number of finite models of classes of models closed under substructure.

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