Definable sets, definable functions, types, realizations, saturated models, stability theory, and classification programs.
Definability studies which sets, relations, and functions can be described inside a structure using formulas of a fixed language, and it is one of the main ways model theory connects syntax with the internal geometry of mathematical structures.
The chapter begins with definable sets and functions, where a formula with free variables is used to specify a collection of tuples from a model, and this gives a precise way to ask what the language can express inside that model.
Types are then introduced as consistent collections of formulas describing the possible behavior of an element or tuple, and they allow one to study objects not only by their actual values but also by the logical conditions they satisfy.
The chapter next studies realizations of types, where a type is realized if there is an element or tuple in a model satisfying all formulas in the type, and this distinction between realized and omitted types becomes important in understanding the size and richness of models.
Saturated models are introduced as models that realize many types, and they provide highly complete environments in which logical possibilities are represented by actual elements whenever the size constraints allow it.
The final sections give an overview of stability theory and classification programs, where theories are analyzed according to the behavior of their definable sets and types, leading to a systematic way of comparing different mathematical structures.