A theory is model-complete when every embedding between its models is elementary. Every model-complete theory is inductive and hence admits a forall-exists axiomatization.
A model companion of is a model-complete theory in the same language with . Equivalently each model of either theory embeds into a model of the other. Alternating these embeddings and using the elementary chain theorem proves uniqueness up to logical equivalence. A model companion need not be a syntactic extension of its original theory.
This is the complete first-order theory of real closed fields.
The theory of real closed fields is complete and model-complete. In the language of ordered rings it eliminates quantifiers; in the pure language of rings its definable order prevents quantifier elimination.
In the language of ordered rings, the Theory of real closed fields has quantifier elimination. In its pure ring reduct the order is definable by nonzero squares, so model completeness survives, although quantifier elimination does not. Keeping these two language conventions separate is essential when constructing a model companion.
A non-Archimedean real closed field contains a positive element larger than every standard integer. Compactness constructs one by adjoining a constant and the formulas for all natural numbers .
Model completeness of real closed fields transfers positivity of a rational function from to real closed extensions of . If the function were not a sum of squares, the Artin-Schreier ordering criterion would produce a real closure in which it is negative, giving a contradiction.
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In mathematical logic, a **model complete theory** is a type of first-order theory that has a specific structure regarding its models. A theory \( T \) is called model complete if every embedding (i.e., a structure-preserving map) between any two models of \( T \) is an isomorphism when the models are elementarily equivalent.