A Kripke model for a first order language is called a partially-elementary extension model if its accessibility relation is not merely a submodel relation but a stronger relation of being an elementary submodel with respect to some class of fromulae. As a main result of the paper, we give a characterization of partially-elementary extension Kripke models. Throughout the paper we exploit a generalized version of the hierarchy of first order formulae introduced by W. Burr. We present some applications of partially-elementary (...) extension Kripke models to subtheories of Heyting Arithmetic and provide examples of their models and prove some of their properties. For example, we show that finite models of subtheories in question need not be normal . The presented results show that the properties of models of subtheories of Heyting Arithmetic differ much from the properties of models of the full theory. (shrink)

From classical, Fraïissé-homogeneous, ($\leq \omega$)-categorical theories over finite relational languages, we construct intuitionistic theories that are complete, prove negations of classical tautologies, and admit quantifier elimination. We also determine the intuitionistic universal fragments of these theories.

We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in (...) the subcategory is mapped to a classical submodel of the corresponding classical model in the range of the original Kripke model. We call a sentence universal if it is built inductively from atoms using ∧, ∨, ∀, and →, with the restriction that antecedents of → must be atomic. We prove that an intuitionistic theory is axiomatized by universal sentences if and only if it is preserved under Kripke submodels. We also prove the following analogue of a classical model-consistency theorem: The universal fragment of a theory Γ is contained in the universal fragment of a theory Δ if and only if every rooted Kripke model of Δ is strongly equivalent to a submodel of a rooted Kripke model of Γ. Our notions of Kripke submodel and universal sentence are natural in the sense that in the presence of the rule of excluded middle, they collapse to the classical notions of submodel and universal sentence. (shrink)

We define classes Φnof formulae of first-order arithmetic with the following properties:(i) Everyφϵ Φnis classically equivalent to a Πn-formula (n≠ 1, Φ1:= Σ1).(ii)(iii)IΠnandiΦn(i.e., Heyting arithmetic with induction schema restricted to Φn-formulae) prove the same Π2-formulae.We further generalize a result by Visser and Wehmeier. namely that prenex induction within intuitionistic arithmetic is rather weak: After closing Φnboth under existential and universal quantification (we call these classes Θn) the corresponding theoriesiΘnstill prove the same Π2-formulae. In a second part we consideriΔ0plus collection-principles. We (...) show that both the provably recursive functions and the provably total functions ofare polynomially bounded. Furthermore we show that the contrapositive of the collection-schema gives rise to instances of the law of excluded middle and hence. (shrink)

We study the relations of being substructure and elementary substructure between Kripke models of intuitionistic predicate logic with the same arbitrary frame. We prove analogues of Tarski's test and Löwenheim-Skolem's theorems as determined by our definitions. The relations between corresponding worlds of two Kripke models [MATHEMATICAL SCRIPT CAPITAL K] ⪯ [MATHEMATICAL SCRIPT CAPITAL K]′ are studied.