Editing Model theory (section)

===Realising and omitting types===
Constructing models that realise certain types and do not realise others is an important task in model theory.
Not realising a type is referred to as ''omitting'' it, and is generally possible by the ''(Countable) Omitting types theorem'':

:Let <math>\mathcal{T}</math> be a theory in a countable signature and let <math>\Phi</math> be a countable set of non-isolated types over the empty set.
:Then there is a model <math>\mathcal{M}</math> of <math>\mathcal{T}</math> which omits every type in <math>\Phi</math>.{{sfn|Hodges|1993|p=333}}

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:

:Let <math>\mathcal{M}</math> be a structure and let <math>\Phi</math> be a set of complete types over a given parameter set <math>A \subset \mathcal{M}.</math>
:Then there is an elementary extension <math>\mathcal{N}</math> of <math>\mathcal{M}</math> which realises every type in <math>\Phi</math>.{{sfn|Hodges|1993|p=451}}

However, since the parameter set is fixed and there is no mention here of the cardinality of <math>\mathcal{N}</math>, this does not imply that every theory has a saturated model.
In fact, whether every theory has a saturated model is independent of the axioms of [[Zermelo–Fraenkel set theory]], and is true if the [[generalised continuum hypothesis]] holds.{{sfn|Hodges|1993|p=492}}