# Model theory

"if proof theory is about the sacred, then model theory is about the profane"

The most prominent scholarly organization in the field of model theory is the Association for Symbolic Logic.

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 the subject, and the two directions are summarised by the pithy characterisations from 1973 and 1997 respectively:

where universal algebra stands for mathematical structures and logic for logical theories; and

Nonetheless, the interplay of classes of models and the sets definable in them has been crucial to the development of model theory throughout its history. For instance, while stability was originally introduced to classify theories by their numbers of models in a given cardinality, stability theory proved crucial to understanding the geometry of definable sets.

Similarly, if σ' is a signature that extends another signature σ, then a complete σ'-theory can be restricted to σ by intersecting the set of its sentences with the set of σ-formulas. Conversely, a complete σ-theory can be regarded as a σ'-theory, and one can extend it (in more than one way) to a complete σ'-theory. The terms reduct and expansion are sometimes applied to this relation as well.

Since we can negate this formula, every cofinite subset (which includes all but finitely many elements of the domain) is also always definable.

While not every type is realised in every structure, every structure realises its isolated types. If the only types over the empty set that are realised in a structure are the isolated types, then the structure is called atomic.

This implies that if a theory in a countable signature has only countably many types over the empty set, then this theory has an atomic model.

On the other hand, there is always an elementary extension in which any set of types over a fixed parameter set is realised:

In particular, any ultraproduct of models of a theory is itself a model of that theory, and thus if two models have isomorphic ultrapowers, they are elementarily equivalent. The Keisler-Shelah theorem provides a converse:

Morley's proof revealed deep connections between uncountable categoricity and the internal structure of the models, which became the starting point of classification theory and stability theory. Uncountably categorical theories are from many points of view the most well-behaved theories. In particular, complete strongly minimal theories are uncountably categorical. This shows that the theory of algebraically closed fields of a given characteristic is uncountably categorical, with the transcendence degree of the field determining its isomorphism type.

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

Model-theoretic results have been generalised beyond elementary classes, that is, classes axiomatisable by a first-order theory.