A first-order theory is complete if for every sentence in its language, either the sentence or its negation is entailed by the theory; equivalently, all models of the theory are elementarily equivalent and the theory decides all sentences of the language.

Principle

Completeness of a theory removes propositional undecision within its language: no sentence is left undetermined by the axioms, so the theory partitions the space of sentences into those it entails and those it refutes, providing maximal syntactic determination given the language.

Demonstration

The theory of dense linear orders without endpoints is complete: any sentence in the language of orders is either provable or refutable from the axioms, so all countable dense linear orders without endpoints are elementarily equivalent. Another example is the complete theory of algebraically closed fields of fixed characteristic and fixed transcendence degree predicates (when parameters are controlled).

Misapplication

Mistaking completeness of a theory for decidability or for model-theoretic saturation: a theory can be complete but undecidable (no algorithm to decide membership), and completeness does not guarantee that models realize all possible types (saturation); conflating these leads to incorrect expectations about computation or model richness.

Consequence

A complete theory gives a sharp classification of sentences and implies that any two models satisfying the theory satisfy exactly the same first-order sentences; this enables clear transfer of first-order properties and simplifies classification problems centered on elementary equivalence.

Reversal

An incomplete theory leaves sentences neither provable nor refutable, producing nontrivial distinctions among models (non-elementary equivalence) and enabling independent extensions or differing completions; incompleteness is a resource for constructing diverse models and capturing underspecified phenomena.

Boundary

Completeness is relative to a chosen language and set of axioms: enlarging the language or weakening/strengthening axioms can break or create completeness; it is strictly syntactic/semantic in first-order logic and does not by itself address higher-order expressivity or meta-theoretical decidability.

Semantic Tension

Tension exists between completeness and decidability (syntactic algorithmic check): completeness ensures every sentence is decided semantically by the theory but does not guarantee there is an effective procedure to decide membership; also completeness can conflict with flexibility needed for certain model constructions.

Synthesis

A complete theory is one that leaves no first-order sentence undecided: it yields uniform elementary behaviour across its models and gives maximal syntactic determination within the chosen language while remaining distinct from algorithmic decidability and model saturation properties.