Semantic proof of the craig interpolation theorem for intuitionistic logic and extensions. part I

This chapter describes the semantic proof of the Craig interpolation theorem for intuitionistic logic and extensions. A semantic proof of a form of Robinson's consistency theorem for the intuitionistic predicate calculus is given. This form is equivalent to the Craig interpolation theorem, for any extension of the intuitionistic predicate calculus. The chapter constructs two partially ordered sets of saturated theories that are isomorphic in a certain sense because in the intuitionistic case, a model is obtained not from a single theory but from a partially ordered set of saturated theories. Two structures, one associated with the $-theories and other associated with the ࡠ-theories, are constructed; it is proved that these two structures are isomorphic in the common language L₀ M₀.