information:extraction
Consistency and completeness

A syntactically consistent theory is a theory from which not every sentence in the underlying language can be proven (with respect to some deductive system, which is usually clear from context). In a deductive system (such as first-order logic) that satisfies the principle of explosion, this is equivalent to requiring that there is no sentence φ such that both φ and its negation can be proven from the theory.

A satisfiable theory is a theory that has a model. This means there is a structure M that satisfies every sentence in the theory. Any satisfiable theory is syntactically consistent, because the structure satisfying the theory will satisfy exactly one of φ and the negation of φ, for each sentence φ.

A consistent theory is sometimes defined to be a syntactically consistent theory, and sometimes defined to be a satisfiable theory. For first-order logic, the most important case, it follows from the completeness theorem that the two meanings coincide. In other logics, such as second-order logic, there are syntactically consistent theories that are not satisfiable, such as ω-inconsistent theories.

A complete consistent theory (or just a complete theory) is a consistent theory T {\displaystyle {\mathcal {T}}} such that for every sentence φ in its language, either φ is provable from T {\displaystyle {\mathcal {T}}} or T {\displaystyle {\mathcal {T}}} ∪ {\displaystyle \cup } {φ} is inconsistent. For theories closed under logical consequence, this means that for every sentence φ, either φ or its negation is contained in the theory. An incomplete theory is a consistent theory that is not complete.

(see also ω-consistent theory for a stronger notion of consistency.)