modeltheoretische

Modeltheoretische (model-theoretic) refers to aspects of model theory, a branch of mathematical logic dealing with the relationship between formal languages and their interpretations in mathematical structures.

Model theory studies how formal languages are interpreted as structures, called models. A theory is a set of sentences; a model of a theory is a structure in which all sentences hold. The central notions include satisfaction, elementary equivalence, completeness, consistency, and the idea of definable sets within a model.

Core results and concepts include the compactness theorem and Löwenheim–Skolem theorems, which relate the sizes of models to the theories they satisfy. Quantifier elimination and model completeness describe theories with particularly tame definable sets; for example, the theories of algebraically closed fields and real closed fields are complete and, in many cases, admit quantifier elimination.

Methods used in model-theoretic practice include Ehrenfeucht–Fraïssé games to compare structures, as well as the development of types, stability theory, and o-minimality to study tameness and structural behavior of models. These tools help classify theories and understand the geometry of definable sets.

Applications of modelltheoretische methods appear across mathematics, notably in algebra, algebraic geometry, and number theory. Model theory provides transfer principles, describes the definable landscape within various objects (fields, groups, valued fields), and offers a framework for comparing mathematical structures through a semantic–syntactic lens.