Model Companion and Model Completion
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.
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:
- T* is a model completion of T
- T has the amalgamation property.
If T also has universal axiomatization, both of the above are also equivalent to:
- T* has elimination of quantifiers
Read more about this topic: Model Complete Theory
Famous quotes containing the words model and/or companion:
“The playing adult steps sideward into another reality; the playing child advances forward to new stages of mastery....Child’s play is the infantile form of the human ability to deal with experience by creating model situations and to master reality by experiment and planning.”
—Erik H. Erikson (20th century)
“The Indian remarked as before, “Must have hard wood to cook moose-meat,” as if that were a maxim, and proceeded to get it. My companion cooked some in California fashion, winding a long string of the meat round a stick and slowly turning it in his hand before the fire. It was very good. But the Indian, not approving of the mode, or because he was not allowed to cook it his own way, would not taste it.”
—Henry David Thoreau (1817–1862)