This page focuses on finitary first order model theory of infinite structures.
The relative emphasis placed on the class of models of a theory as opposed to the class of definable sets within a model fluctuated in the history of

 ...

First-order logic
A first-order formula is built out of atomic formulas such as R(f(x,y),z) or y = x + 1 by means of the Boolean connectives and prefixing of quantifiers or . A sentence is a formula in which each occurrence of a variable is in the scope of

 ...

Definable sets
In model theory, definable sets are important objects of study. For instance, in the formula

 ...

Basic notions
For a sequence of elements of a structure and a subset A of , one can consider the set of all first-order formulas with parameters in A that are satisfied by . This is called the complete (n-)type realised by over A. If there is an

 ...

Realising and omitting types
Constructing models that realise certain types and do not realise others is an important task in model

 ...

A theory was originally called categorical if it determines a structure up to isomorphism. It turns out that this definition is not useful, due to serious restrictions in the

 ...

A key factor in the structure of the class of models of a first-order theory is its place in the stability hierarchy.
A complete theory T is

 ...