A property of a first-order theory T saying that every embedding between models of T is an elementary embedding; equivalently, every formula is T-equivalent to an existential formula (or a universal formula after negation) so truth is preserved under substructure embeddings that are embeddings of models of T.

Principle

Reduce arbitrary formula truth to existential assertions: if embeddings between models are elementary then syntactic complexity can be collapsed to existential form, allowing transfer of definable properties across embeddings.

Demonstration

The theory of algebraically closed fields of fixed characteristic is model-complete: any embedding of one algebraically closed field into another is elementary, and many definable conditions can be expressed by existential polynomial equations.

Misapplication

Treating a model-complete theory as if it had full quantifier elimination; model-completeness does not imply every formula is quantifier-free, so assuming quantifier-free classification or eliminating quantifiers when they cannot be eliminated is incorrect.

Consequence

One obtains a robust preservation theorem: existential consequences control extension behaviour, model companions (when they exist) are often model-complete, and many transfer arguments reduce to checking existential formulas.

Reversal

A model-incomplete theory admits embeddings between models that are not elementary; some formulas change truth value under embeddings, so existential formulas do not suffice to capture all definable properties.

Boundary

Applies to first-order theories in a fixed language; model-completeness is a syntactic/semantic property of theories, not of arbitrary classes of structures outside first-order logic or without the specified language.

Semantic Tension

Close to quantifier elimination but strictly weaker: quantifier elimination forces equivalence to quantifier-free formulas while model-completeness allows equivalence to existential formulas only; also distinct from completeness of the theory itself.

Synthesis

Model completeness is the condition that embeddings preserve all first-order truths and that the content of the theory can be checked through existential descriptions; it streamlines model-theoretic arguments without demanding full quantifier elimination.