indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
Existential closedness is a property of first-order structures (in the same way that model completeness is a property of first-order theories) meant to generalize the properties of the theory ACF of algebraically closed fields.
As discussed in section 8.1 of (Hodges93):
Take a first-order language without relation symbols and a class of -structures. For example, might be the language of rings and the class of fields, or might be the language of groups and the class of groups.
We say that a structure in is existentially closed in if whenever is a finite set of equations and inequations with parameters from and has a simultaneous solution in some extension of with in , then has a solution already in .
(c.f. algebraically closed field).
An existential formula is a formula of the form where is quantifier-free.
A universal formula is a formula of the form where is quantifier-free.
We say that a model of a first-order theory is existentially closed if for every also modelling , every embedding , every tuple , and every existential formula , then whenever , we also have that .
(“The existence of witnesses in a superstructure to existential statements about points in the existentially closed model is reflected to the existentially closed model.”)
In the terminology of the passage from (Hodges93), we say that is an existentially closed model of if it is existentially closed in the class .
We say that a theory is an “existentially closed theory” (i.e., model complete) if all of its models are existentially closed.
The theory ACF of algebraically closed fields is existentially closed. An existential formula in says “for a choice of coefficients , there exists a point in affine space (of fixed dimension) which is in these finitely many affine varieties depending on but not these finitely many affine varieties depending on ”.
Checking that is existentially closed therefore means checking that if is an embedding of algebraically closed fields, and is a choice of coefficients from , then if such a point exists in affine -space, then we can find a similar in affine -space.
However, we can do so very cheaply after we know that eliminates quantifiers (and therefore so does the theory of algebraically closed fields containing ): is equivalent to some quantifier-free sentence involving , and quantifier-free sentences are always transferred by embeddings of structures, so we are done.
model companion?
Last revised on May 23, 2017 at 03:45:34. See the history of this page for a list of all contributions to it.