Atomic model (mathematical logic)

An atomic model is in the model theory, a branch of mathematical logic a model that therefore in a certain sense is realized only very few types and very small. The term is related to the concept of Primmodells and is dual to the notion of the saturated model, which in turn realized many types.

Definition

The considered languages ​​are hereinafter always at most countable.

• A guy on a theory is an atomic type if there is a formula, so that applies to all:

• A model of a theory is atomic when it realized only atomic types, so if for each of the type: is atomic over.

Since finite subsets of types are always realized, atomic ( isolated ) types are always realized. For countable languages ​​of the Omitting Types - set says that exactly are the types, which are always realized.

Examples

• The theory of real closed fields as atomic model the field of real algebraic numbers.
• Every finite model is atomic.
• Each model in the empty language is atomic. In particular, this example provides uncountable atomic models.
• Each model in the theory of dense linear order without endpoints is atomic.

Existence

We have the following sentence:

• A complete theory has exactly then an atomic model when the isolated ( atomic ) types are close in their lie.

Characterization

• Countable atomic models of a complete theory are isomorphic.
• A model of a complete theory is accurate then a countable atomic model if it is a prime model.

• If a complete theory has a countable saturated model, it also has atomic model.

Never two

An application of the theory of atomic and saturated models is the following proven by Vaught sentence:

• No complete theory (over a countable language ) has exactly two nonisomorphic countable models.

In the proof we used the fact that in exactly two non- isomorphic models of an atomic () and the other ( ) should be saturated. Realized now a non- insulated types, so if one considers the theory of. With the theory of atomic and saturated models one can conclude then, that there must be an atomic model of this theory and that the reduced model of this theory is isomorphic to neither may still be.

For example, the theory about the language has exactly three countable models. is the atomic model. The saturated model is the model. The model between these two. ( The constant symbols are always carried the natural numbers of the first copy of interpreted ). In the last model the constant sequence has a supremum, it is neither atomic nor saturated.

The theory can be explained by the addition of single-digit predicate symbols expand so that it has exactly models: the axioms are expanded by the sentences that just applies to each element of a predicate, that the constants satisfy the predicate and that are both tight. There is then an atomic and a saturated model. In addition, models exist where the constant sequence has a supremum, but each meet a different predicate.