We prove that an intermediate predicate logic characterized by a class of finite partially ordered sets is recursively axiomatizable iff it is “finite”, i.e., iff it is characterized by a single finite partially ordered set. Therefore, the predicate logic LFin of the class of all predicate Kripke frames with finitely many possible worlds is not recursively axiomatizable.
"The superintuitionistic predicate logic of finite Kripke frames is not recursively axiomatizable." J. Symbolic Logic 70 (2) 451 - 459, June 2005. https://doi.org/10.2178/jsl/1120224722