Constructive Kripke Semantics and Realizability

What is the truth-value structure of realizability? How can realizability style models be integrated with forcing techniques from Kripke and Beth semantics, and conversely? These questions have received answers in Hyland’s [33], Läuchli’s [43] and in other, related or more syntactic developments cited below. Here we re-open the investigation with the aim of providing more constructive answers to both questions. A special, constructive class of so-called fallible Beth models has been shown intuitionistically to be complete for intuitionistic logic by Friedman, Veldman and others. Here we build such intuitionistic Beth models elementarily equivalent to a natural and broad class of realizabilities, thus showing that realizability interpretations correspond to the particularly effective kind of models yielded by the Veldman – Friedman – de Swart completeness theorem. In the other direction, an abstract realizability is shown constructively to be complete for intuitionistic logic. This extends earlier results along these lines due to Läuchli and others.