A first-order theory T in a language L is complete if for every L-sentence φ either T proves φ or T proves ¬φ; equivalently, any two models of T are elementarily equivalent (T decides every sentence of the language).

Principle

Completeness expresses that the theory assigns a definite first-order truth-value to every sentence of its language, so syntactic deduction and semantic truth coincide at the level of sentences for that theory.

Demonstration

The theory of dense linear orders without endpoints (DLO) is complete: every sentence in the language of linear orders is either provable from DLO or its negation is provable, and all countable dense orders without endpoints are isomorphic to (Q,<), so DLO decides first-order sentences about order-denseness and endpoints.

Misapplication

Confusing completeness with decidability (a complete theory may be undecidable) or with model-completeness (a complete theory need not be model-complete); assuming completeness transfers automatically across expansions of the language without care.

Consequence

A complete theory yields a fixed first-order picture of its models: types over the empty set are determined, and any two models satisfy exactly the same sentences, facilitating classification and transfer of properties between models.

Reversal

An incomplete theory leaves some sentences independent (neither provable nor refutable), allowing multiple non-elementarily-equivalent models and genuine ambiguity in first-order truths about the language.

Boundary

Completeness is relative to the chosen language and signature: adding symbols can break completeness; completeness does not imply stronger metatheoretic properties like decidability or categoricity without extra hypotheses.

Semantic Tension

Tension with model-completeness and decidability: model-complete theories have different embedding behavior than complete ones, and decidability concerns effective determination of provability, a distinct notion from syntactic completeness.

Synthesis

A complete theory is a first-order theory that decides every sentence in its language, giving a determinate semantic profile to all its models though not necessarily yielding algorithmic decidability or uniqueness of models in each cardinality.