The Constructive Theory of Riesz Spaces and Applications in Mathematical Economics A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy in Mathematics at the University of Canterbury by Marian Alexandru Baroni Supervisor: Professor Douglas Bridges University of Canterbury Department of Mathematics and Statistics 2004 PHYSICAl. SCIl'NC!:S LIBRARY QA 322 iii Abstract This thesis is an introduction to a constructive development of the theory of or­ dered vector spaces. Order structures are examined constructively; that is, with intuitionistic logic. Since the least-upper-bound principle does not hold constructively, some prob­ lems that are classically trivial are much more difficult from a constructive stand­ point. The first problem in a constructive development of a theory is to find ap­ propriate counterparts of the classical notions. We introduce a positive definition of an ordered vector space and we extend the constructive notions of supremum, or­ der 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 con­ structive 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 relation­ ship 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 func­ tion. 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. iv v Acknowledgements First of all, I would like to thank Professor Douglas Bridges, my supervisor, for his guidance and support. This research would not have been possible without the financial support offered by the University of Canterbury and its Department of Mathematics and Statistics. I would also like to thank the New Zealand Mathematical Society and the New Zealand Mathematics Research Institute for conference and travel support. I am most grateful to Professor Cristian Calude for his inspiring advice and continuous encouragement. Special thanks are due to my professors from the University of Bucharest, Ro­ mania: to Professor Romulus Cristescu who introduced me in the realm of ordered vector spaces, and to Professor Solomon Marcus who encouraged me to undertake this doctoral research. I had a great time at the phylogenetics meetings in Kaikoura and Whakapapa. Warm thanks to Professor Mike Steel and Dr Charles Semple. Thanks to all the people in the Department of Mathematics and Statistics for providing a friendly environment. I am very grateful to Neville and Ella Bolsover, Douglas and Vivien Bridges, and all the friends who made me and my family feel at home in New Zealand. A word of gratitude to my friends in Romania-Cristina Costache, Elena Costache, and CataJina Iticescu-for their kind and efficient help. I wish to thank my family: my wife, Mihaela, for her patience during three years of sharing the same office, and my daughter, Andreea, who still wonders what this thesis is useful for. I am forever grateful to my parents, Marian and Virginia Baroni, for their love and support over the years. This thesis is dedicated to their memory. vi Contents 1 Introduction 1 1.1 Intuitionistic logic . 1 1.2 Bishop's constructive mathematics 3 1.3 Sources of nonconstructivity 5 1.4 An example . 6 1.5 An overview of the thesis . 8 2 Ordered vector spaces 11 2.1 Preliminaries ............................... 11 2.2 Partially ordered sets .......................... 12 2.3 Ordered vector spaces 15 2.4 Suprema and infima ........................... 21 2.5 Lattices . .. 28 2.6 Riesz spaces . 31 3 Dedekind complete sets 35 viii CONTENTS 3.1 Order locatedness . .. 36 3.2 Dedekind completeness. .. 38 3.3 The product order ............................ 40 3.4 An example: Rn ............................. 43 4 Order units 49 4.1 Archimedean spaces. 49 4.2 Lattice norms . .. 52 4.3 The Minkowski functional associated with an order unit. .. 54 4.4 Krein spaces . 60 5 Positive operators 63 5.1 Monotonicity . .. 64 5.2 Positivity .. .. 66 5.3 Order bounded operators. .. 70 5.4 The order dual .... .. 74 6 Applications in mathematical economics 77 6.1 Preference and utility: an introduction . .. 77 6.2 Basic definitions. .. 78 6.3 Strong extensionality . .. 79 6.4 Continuity................................. 82 CONTENTS IX 6.5 Monotone weak excess relations . .. 87 6.6 Unit elements . ., 94 6.7 Uniformly proper preferences. .. 99 Bibliography 101 List of Symbols 109 Index 111 x CONTENTS Chapter 1 Introduction This work is a first step towards a constructive development of the theory of or­ dered vector spaces. Almost every vector space over the real field has a natural ordering that is compatible in a certain way with the algebraic structure of the space. However, although the classical theory has been considerably developed in the last decades, constructive mathematics has paid much less attention than its classical counterpart to the order structures of vector spaces. Before starting our constructive examination of ordered vector spaces, we should clarify its setting. By constructive mathematics, we mean Bishop-style mathematics which enables one to interpret the results both in classical mathematics and in other varieties of constructivism. We adopt Fred Richman's viewpoint [60]: constructive mathematics is simply mathematics carried out with intuitionistic logic. 1.1 Intuitionistic logic When is a mathematical statement accepted as valid? For the traditional point of view, a statement is true whenever is not contradictory. Although certain mathe­ maticians had expressed dissatisfaction regarding the idealistic content of mathemat­ ics, nonconstructive methods became standard at the end of the nineteenth century. 2 Introduction It was the Dutch mathematician Luitzen Egbertus Jan Brouwer (1881-1966) who attempted a fully constructive approach to mathematics. Brouwer's views about mathematics were presented first in his 1907 doctoral thesis [25]. In Brouwer's phi­ losophy, known as "intuitionism", mathematics is a free creation of the human mind, and an object exists only if it can be (mentally) constructed. When we work constructively, the logical connectives and quantifiers should be interpreted in a different manner. • To prove P 1\ Q, we need a proof of P and a proof of Q. • To prove P V Q, we must have either a proof of P or a proof of Q. Unlike in classical logic, it is not enough to prove the impossibility of ....,p 1\ ....,Q. • To prove P =} Q, we have to provide an algorithm that converts a proof of P into a proof of Q. 1 • To prove ....,P, we must show that P implies 0 = 1. • To prove :lx E A P(x), we require an algorithm that produces an object x and a proof that P(x) holds. • To prove \Ix E A P(x), we need an algorithm that, applied to an object x and a proof that x E A, proves that P(x) holds. In view of the above interpretation of disjunction, the universal validity of the Aristotelian principle Tertium non datur (the law of excluded middle), PV....,P, should be questioned. (We shall return to this subject m Section 1.3.) It was Brouwer [26] the first who observed that the law of excluded middle was extended without justification to statements about infinite sets. 1 A proof of P is not required. However, if P has a proof, we should be able to obtain a proof of Q. 1.2 Bishop's constructive mathematics 3 The analysis of the logical principles used in constructive proofs led Heyting2 to the axioms of intuitionistic logic. Both Brouwer's intuitionistic mathematics and another variety of constructivism, the recursive constructive mathematics initiated by A.A. Markov, are based on intuitionistic logic together with some additional principles. 3 On the other hand, classical logic can be obtained from the axioms of intuitionistic logic by adding the law of excluded middle. 1.2 Bishop's constructive mathematics In the 1960s, although intuitionism and Markov's constructive mathematics had inspired much work in logic and metamathematics, it was not at all clear that classical mathematics had a satisfactory alternative. The common viewpoint was still that of Hilbert [35]: Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists. The publication of the monograph Foundations of Constructive Analysis [8] by Errett Bishop (1928-1983) changed the situation dramatically. As observed in one of the reviews [68] of the book, Bishop showed "that to replace the classical system by the constructive one does not in any way mutilate the great classical theories of mathematics. Not at all. If anything, it strenghthens them, and shows them, in a truer light, to be far grander than we had known." Bishop showed that, contrary to the common view, a constructive development of a large part of analysis was possible. Not only was the content important but also the mathematics was written in a normal mathematical style.