Model complete theory

In model theory, a branch of mathematical logic, ie, a theory model complete if sub-models are particularly good at their top model.

Definition

A theory is called model complete if for two models and from that follows that is in elementary, in characters.

Robinsons test

To prove the model completeness can be used frequently Robinsons test. A formula of a language is called existential if it of the form

Is quantifier-free. The analogy a universal formula, if it of the form

Is quantifier-free. If two models, so called existentially closed in, if any existential statement of the language that applies, in true.

Robinsons test is:

For a set of propositions is equivalent to:

Completeness versus model completeness

A complete theory need not be model- complete nor has a model- complete theory to be complete. Where, however, a model- complete theory, a model that can be embedded in any other model of the theory, this theory is also complete. (see prime model )

Model companion

A theory is called model companion of a theory, if

• Can be extended by a model of each model and
• Is model complete.

It can be shown that for each theory exists at most one model companion.

Examples

• The theory of dense linear open total order is complete and model complete. She is the model companion of the theory of linear orders.
• The theory of algebraically closed body ( without knowledge of the characteristic) is not complete, but the model completely.
• The theory of algebraically closed fields of a fixed characteristic has a prime model and is both fully model complete.
• The theory of dense linear order with Total extremes is complete but not model- complete. The interval is not elementary in the interval.