model

:

• in #model theory, a model of a theory is a realization of the types, operations, relations, and axioms of that theory
• In ordinary model theory one usually studies mainly models in sets, but in categorical logic we study models in other categories, especially in topoi.
• The term structure is often used to mean a realization of types, operations, and relations in some signature, but not satisfying any particular axioms.
  • ie, the same as a model for the “empty theory” in that signature, which has same types, operations, and relations, but no axioms
  • One then talks about whether a given structure is, or is not, a model of a given theory in a given signature.

model theory

• On the one hand, there is syntax. On the other hand, there is semantics. Model theory is (roughly) about the relations between the two: model theory studies classes of models of theories, hence classes of “mathematical structures”.