Editing Soundness (section)

===Relation to completeness===

The converse of the soundness property is the semantic [[Completeness (logic)|completeness]] property. A deductive system with a semantic theory is strongly complete if every sentence ''P'' that is a [[semantic consequence]] of a set of sentences Γ can be derived in the [[deduction system]] from that set. In symbols: whenever {{nowrap|Γ <big>⊨</big> ''P''}}, then also {{nowrap|Γ <big>⊢</big> ''P''}}. Completeness of [[first-order logic]] was first [[Gödel's completeness theorem|explicitly established]] by [[Gödel]], though some of the main results were contained in earlier work of [[Skolem]].

Informally, a soundness theorem for a deductive system expresses that all provable sentences are true. Completeness states that all true sentences are provable.

[[Gödel's incompleteness theorem|Gödel's first incompleteness theorem]] shows that for languages sufficient for doing a certain amount of arithmetic, there can be no consistent and effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language. Thus, not all sound deductive systems are complete in this special sense of completeness, in which the class of models (up to [[isomorphism]]) is restricted to the intended one. The original completeness proof applies to ''all'' classical models, not some special proper subclass of intended ones.