Submodels of Kripke Models
Albert Visser · 1998
A Kripke model K is a submodel of another Kripke modelMif K is obtained by restricting the set of nodes of M. In this paper we showthat the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. This result is an analogue of the Los-Tarski theorem for the Classical Predicate Calculus. In appendix A we provethat for theories with decidable identity we can take as the embeddings between domains in Kripke models of the theory, the identical embeddings. This is a well known fact, but we know of no correct proof in the literature. In appendix B we answer, negatively, a question posed by Sam Buss: whether there is a classical theory T, such that HT is HA. HereHT is the theory of all Kripke modelsMsuch that the structures assigned to the nodes of M all satisfy T in the sense of