Quantifier Elimination for a Class of Intuitionistic Theories

From classical, Fräıssé-homogeneous, (≤ ω)-categorical theories over finite relational languages (which we refer to as JRS theories), we construct intuitionistic theories that are complete, prove negations of classical tautologies, and admit quantifier elimination. The technique we use considers Kripke models as functors from a small category to the category of L-structures with morphisms, rather than the usual interpretation wherein the frame of a Kripke model is a partial order. While one can always “unravel” a functor Kripke model to obtain a partial order Kripke model with the same intuitionistic theory, our technique is perhaps an easier way to consider a Kripke model that includes a single classical node structure and all of the endomorphisms of that classical JRS structure. We also determine the intuitionistic universal fragments of these theories, in accordance with the hierarchy of intuitionistic formulas put forth in [9] and expounded on by Fleischmann in [11]. This portion of the thesis (Chapter 1) is the result of joint work with Ben Ellison, Jonathan Fleischmann, and Wim Ruitenburg, as published (up to minor structural changes) in [10]. Given a classical JRS theory, we determine axiomatizations of the corresponding intuitionistic theory in Chapter 2. We first do so by axiomatizing properties apparent from the behavior of the model, and discuss improvements to that axiom system. We then present another axiomatization, this time by axiomatizing the properties of quantifier elimination. We discuss improvements to this system, and show how this system and various subsystems thereof are equivalent to our first axiomatization and corresponding subsystems thereof.