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. This is of course the same as a model for the “empty theory” in that signature, which has the same types, operations, and relations, but no axioms at all. One then talks about whether a given structure is, or is not, a model of a given theory in a given signature.
An -structure is a model of if for every sentence in , its interpretation in , is true (” holds in ”).
We say that is consistent or satisfiable (relative to the universe in which we do model theory) if there exist at least one model for (in our universe). Two theories, , are said to be equivalent if they have the same models.
Given a class of structures for , there is a theory consisting of all sentences in which hold in every structure from . Two structures and are elementary equivalent (sometimes written by equality , sometimes said “elementarily equivalent”) if , i.e. if they satisfy the same sentences in . Any set of sentences which is equivalent to is called a set of axioms of . A theory is said to be finitely axiomatizable if there exist a finite set of axioms for .
A theory is said to be complete if it is equivalent to for some structure .