Prime model

In model theory, a branch of mathematical logic is called a model of a theory prime model if this model can be embedded in each elementary model of this theory.

Definition

Below is a countable theory with no finite models.

Is a prime model of the theory iff for all with a picture exists with

Examples

• The algebraic closure of the prime field (or ) is a prime model of the theory of algebraically closed field of characteristic (or 0 ).
• Is a prime model of dense linear orders without extremes.

Properties

• It follows from the Löwenheim - Skolem that a prime model is countable.
• Actual categorically, the countable model is a prime model.
• Two Primmodelle a theory are isomorphic.
• One theory has exactly one prime model then when the isolated types are dense.

Example of a theory with no prime model

The following theory of language does not have a prime model: the language contains, for each one -place predicate.

(For the notation: is the set of all finite sequences consisting of only zeros or ones. )

The axioms of the theory are ( runs through all finite sequences ):

The theory has no isolated types and therefore no prime model.