Type (model theory)

Lore & Background

A 1-type over a set A is a set p(x) of formulas in L(A) with at most one free variable x, such that for every finite subset p0(x) there is some element b in the structure's universe with the structure modeling p0(b). More generally, an n-type is defined similarly for n-tuples. A complete type is maximal with respect to inclusion, meaning for every formula either it or its negation is in the type; any non-complete type is called partial.

Reader's Guide

Types are central to model theory because they capture the complete or partial description of an element or tuple relative to a given set of parameters. The concept of a type being realized in a structure or only in an elementary extension is fundamental. Isolated types, which are implied by a single formula, are always realized in every elementary substructure or extension. The study of which types can be omitted leads to the omitting types theorem, and models that realize the maximum possible variety of types are called saturated models. The ultrapower construction provides one way of producing saturated models.

Did You Know?

• A complete type is maximal with respect to inclusion; for every formula, either it or its negation is in the type.
• Any type is realizable in some elementary extension of the structure, by the compactness theorem.
• An isolated type is one for which there is a single formula that implies all other formulas in the type.
• The only complete 0-type is the set of all sentences of L(A) that are modelled by the structure.