Towards an Algebraic Topos Semantics for Three-valued Gödel Logic

The algebraic semantics of Gödel propositional logic is given by the variety of Gödel algebras, which in turns form a category dually equivalent to the pro-finite completion of the category of finite forests and order-preserving open maps. Forests provide a sound and complete semantics for propositional infinite-valued Gödel logic, while propositional k-valued Gödel logic is sound and complete for forests of height at most k-1. In this work we shall mainly deal with three-valued Gödel logic. We shall show that the subcategory of forests of height at most 2 (bushes) forms an elementary topos, thus providing naturally a generalisation to bushes of all classical first-order set concepts, suitable for developing a first-order three-valued Gödel logic semantics based on bush concepts instead of sets.