The Countable Henkin Principle

This is a revised and extended version of an article which encapsulates a key aspect of the “Henkin method” in a general result about the existence of finitely consistent theories satisfying prescribed closure conditions. This principle can be used to give streamlined proofs of completeness for logical systems, in which inductive Henkin-style constructions are replaced by a demonstration that a certain theory “respects” some class of inference rules. The countable version of the principle has a special role, and is applied here to omitting-types theorems, and to strong completeness proofs for first-order logic, omega-logic, countable fragments of languages with infinite conjunctions, and a propositional logic with probabilistic modalities. The paper concludes with a topological approach to the countable principle, using the Baire Category Theorem. The Henkin method is the technique for constructing maximally consistent theories satisfying prescribed closure conditions that was introduced by Leon Henkin in his 1947 doctoral dissertation. The method involves building up the desired theory by induction along an enumeration of some relevant class of formulas, with choices being made at each inductive step to include certain formulas, in such a way that when the induction is finished the theory has the properties desired. The character of this procedure is neatly captured in a phrase of Sacks [20, p. 30], who attributes its importance to the fact that it “takes into account the ultimate consequences of decisions made at intermediate stages of the construction”. Famously, Henkin used his method to give the first new proof of completeness of first-order logic since the original 1929 proof of Gödel, and to prove completeness of a theory of types with respect to “general” models [12, 13]. He obtained the type theory result first and thought it would be of greater interest, as he explained in the remarkable article [16] in which he tells the story of his “accidental” discovery of these completeness proofs while trying to solve a different problem. In fact logicians have paid more attention to the first-order construction, which was eventually adapted to propositional and first-order versions of modal, temporal, intuitionistic and substructural logics. There are now numerous kinds of