satisfiable having a common model, a structure in which all the sentences in the set are true; said of a set of sentences. In modern logic, satisfiability is the semantic analogue of the syntactic, proof-theoretic notion of consistency, the unprovability of any explicit contradiction. The completeness theorem for first-order logic, that all valid sentences are provable, can be formulated in terms of satisfiability: syntactic consistency implies satisfiability. This theorem does not necessarily hold for extensions of first-order logic. For any sound proof system for secondorder logic there will be an unsatisfiable set of sentences without there being a formal derivation of a contradiction from the set. This follows from Gödel’s incompleteness theorem. One of the central results of model theory for first-order logic concerns satisfiability: the compactness theorem, due to Gödel in 1936, says that if every finite subset of a set of sentences is satisfiable the set itself is satisfiable. It follows immediately from his completeness theorem for first-order logic, and gives a powerful method to prove the consistency of a set of sentences. See also COM- PACTNESS THEOREM , COMPLETENESS , GÖDEL’S INCOMPLETENESS THEOREMS, MODEL THEORY, PROOF THEORY. Z.G.S.