A theory T is complete if for every sentence φ in its language either T proves φ or T proves ¬φ; equivalently, any two models of T are elementarily equivalent, so the theory decides truth or falsity of every sentence up to logical consequence.

Principle

Maximal consistency: a complete theory contains, for each sentence, a definitive choice of truth or negation compatible with its axioms, making it a maximal consistent set of sentences in the language relative to entailment.

Demonstration

Example: the theory of real closed fields in the language of ordered fields is complete: any two real closed fields satisfy the same first-order sentences in that language; similarly, the theory of algebraically closed fields of a fixed characteristic is complete in the language of rings.

Misapplication

Confusing completeness of a theory with decidability (a theory can be complete but undecidable) or with categoricity (a complete theory may still have multiple non-isomorphic models in a given cardinality).

Consequence

Completeness ensures that for any sentence one can in principle determine its provability or refutability from the theory (subject to effectiveness issues), simplifies comparison of models via elementary equivalence, and underpins many classification arguments relying on types and saturation.

Reversal

An incomplete theory leaves some sentences neither provable nor refutable by the axioms; this indeterminacy often reflects genuine model-theoretic diversity among models of the theory.

Boundary

Completeness is relative to the chosen language and deductive notion: changing the language or allowing additional axioms can break completeness; completeness does not itself assert decidability or uniqueness of models of a given size.

Semantic Tension

Completeness vs decidability: completeness is a semantic-syntactic maximality about sentences, whereas decidability demands an effective procedure to determine membership of the theory; a theory can be one without the other.

Synthesis

A complete theory is a maximally informative axiomatization in a language: it decides every sentence (provably or refutably), making all its models elementarily equivalent and providing a firm basis for model-theoretic analysis even when algorithmic questions remain open.