Model complete theory

In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula.This notion was introduced by Abraham Robinson. In model theory, a first-order theory is called model complete if every embedding of its models is an elementary embedding. Equivalently, every first-order formula is equivalent to a universal formula.This notion was introduced by Abraham Robinson. A companion of a theory T is a theory T* such that every model of T can be embedded in a model of T* and vice versa. A model companion of a theory T is a companion of T that is model complete. Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if T {displaystyle T} is an ℵ 0 {displaystyle aleph _{0}} -categorical theory, then it always has a model companion . A model completion for a theory T is a model companion T* such that for any model M of T, the theory of T* together with the diagram of M is complete. Roughly speaking, this means every model of T is embeddable in a model of T* in a unique way. If T* is a model companion of T then the following conditions are equivalent: If T also has universal axiomatization, both of the above are also equivalent to: