*1.1. Model Theory*

In a broader sense, model theory is the mathematical study of language interpretations, its main paradigm being Alfred Tarski's semantic definition of truth [1]. Thus, the occurrence of the symbol |= always indicates that we are in the presence of some form of model–theoretical argument. In its most classical form, model theory deals with *first-order structures*. So, in first-order model theory, the relation *M* |= *ρ* means that *M* is a first-order model and *ρ* is a first-order sentence. Tarski's approach was to determine the validity of this relation inductively on the structure of *ρ*. On the one hand, first-order model theory [2,3] is a vibrant and sophisticated area of mathematical research that brings logical methods to bear on deep problems of classical mathematics. Two early achievements of first-order model theory that brought it fame within the wider mathematical community were the modern and rigorous recovery of the approach to mathematical analysis of Newton, Leibniz and Euler—in the form of Robinson's non-standard analysis [4,5]—and the proof of the independence of the Continuum Hypothesis [6,7]. Moreover, first-order model theory has applications to other scientific areas, most notably to computing science. On the other hand, first-order model theory is the area in which many of the broader ideas of model theory were first worked out.
