Complete Theory

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.

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).

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.