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.