On Constructive Sets and Partial Structures

Abstract: The first three papers in this thesis study the formalisation of a set in type theory as a data type with an equivalence relation – an object usually known as a setoid. The corresponding formalisation of a locally small category is called an E-category.In Paper I, we show that type theory without universes is insufficient for proving that some expected properties hold of the E-category of setoids, but that a minimal universe is sufficient.In Paper II, we show that although the collection of all E-categories does not form a category, we can introduce a type-theoretic version of bicategories, and the E-categories form such an E-bicategory.In Paper III, we consider the setoids inside a type-theoretic universe. The axiom of unique substitutions is proposed and used to show that these form a small category (that is, a category witha setoid of objects and a single setoid of all arrows). We demonstrate that this construction can not be carried out without adding some new axiom to type theory. We also show that the axiom of unique substitutions is strictly weaker than the axiom of unique identity proofs.In Paper IV, we investigate partial equivalence relations, also known as partial setoids, in Heyting arithmetic in all finite types, and adapt the result that the extensional axiom of choice is equivalent to the combination of the intensional axiom of choice, classical logic, and an extensionality axiom.In Paper V, we investigate PHL, a logic of partial terms, and prove a cut elimination theorem for it and for a related calculus.