indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
Model completeness is a property of first-order theories (in the same way that existential closedness is a property of first-order structures) meant to generalize the properties of the theory ACF of algebraically closed fields.
As discussed in Hodges’ Model theory:
In the early 1950s Abraham Robinson noticed that certain maps studied by algebraists are in fact elementary embeddings. If you choose a map at random, the chances of it being an elementary embedding are negligible. So Robinson reckoned that there must be a systematic reason for the appearance of these elementary embeddings, and he set out to find what the reason was.
In the course of this quest he introduced the notions of model complete theories, companionable theories and model companions?. These notions have become essential tools for the model theory of algebra.
We say a first-order theory is model complete if every embedding (not necessarily elementary) between models of is elementary.
The following characterization appears as Theorem 8.3.1 in (Hodges93):
Let be as above. The following are equivalent:
is model complete.
Every model of is existentially closed.
For every embedding where and model , there exists an elementary extension of and an embedding such that the triangle commutes.
Every existential formula is equivalent (mod ) to a universal formula.
Every formula is equivalent (mod ) to a universal formula.
A theory is model complete if and only if for every model of , the quantifier-free diagram (whose models are precisely the models of containing as an embedded substructure) is complete.
Analogously, a theory is substructure complete if and only if for every model of and every substructure , the quantifier-free diagram is complete.
A model complete theory is an “existentially closed theory” in the sense that all of its models are existentially closed models.
As mentioned above, substructure completeness implies model completeness because every elementary submodel is also a substructure. The proposition below shows that the converse holds with some additional assumptions.
Suppose the first-order theory in the language is model complete and has the property that for any two models and a mutual -substructure , the latter span in the category of -structures and embeddings has a cocone which is also a model of .
Then is substructure complete.
This property of being able to “amalgamate” the models and over the common substructure is sometimes called the amalgamation property, though this terminology is rather overloaded (c.f. Fraisse limit).
Let be a substructure of some model . Append the quantifier-free diagram of to to form . Suppose towards a contradiction that is a sentence which is undecided by ; let and be models of which witness this, so that while .
Let amalgamate and over .
By model completeness, the theory of -models (not necessarily elementarily) embedding is complete. Abuse notation and replace the instances of constants from in with their images in . Either or . Since certainly embeds itself and , .
Similarly,
Now, on one hand, the theory (where we glue along the copies of in and ) entails both and , which makes it inconsistent. This theory therefore has no models.
On the other hand, this theory axiomatizes the class of amalgams of and over in the category of -structures and embeddings, so in particular is modelled by , a contradiction.
Last revised on May 27, 2017 at 06:34:03. See the history of this page for a list of all contributions to it.