The constructive theory of Riesz spaces and applications in mathematical economics.
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
This thesis is an introduction to a constructive development of the theory of ordered vector spaces. Order structures are examined constructively; that is, with intuitionistic logic. Since the least-upper-bound principle does not hold constructively, some problems that are classically trivial are much more difficult from a constructive standpoint. The first problem in a constructive development of a theory is to find appropriate counterparts of the classical notions. We introduce a positive definition of an ordered vector space and we extend the constructive notions of supremum, order locatedness, and Dedekind completeness from the real number line to arbitrary partially ordered sets. As a main result, we prove that the supremum of a subset S exists if and only if S is upper located and has a weak supremum-that is, the classical least upper bound. We investigate ordered vector spaces and, in particular, Riesz spaces with order units and their order duals. For an Archimedean space, we obtain several constructive counterparts of a classical theorem that links order units and Minkowski functionals. We also examine linearly ordered vector spaces; it turns out that, as in the classical case, any nontrivial Archimedean space with a linear order is isomorphic to R. Various notions of monotonicity for mappings and for preference relations are discussed. In particular, we examine positive operators and highlight the relationship between strong extensionality and strong positivity-a stronger counterpart of the classical positivity. The last chapter is dedicated to applications in mathematical economics. We deal with the problem of the representation of a preference by a continuous utility function. Since strong extensionality is a necessary condition for such a representation, we examine in detail the relationship between continuity and strong extensionality and we obtain sufficient conditions for the latter property. We apply these results to obtain a theorem of representation for preferences with unit elements.