Zorn's Lemma in rational choice theory: a note

Abstract

The first significant use of Zorn s Lemma in rational choice theory is in the paper by Szpilrajin(1930), where it is established that every strict partial order can be embedded within a strict linear order. Subsequently, Dushnik and Miller [1941] showed that every strict partial order is the intersection of all the strict linear orders in which it can be embedded. Richter ([1966], [1971]), Hansson [1968], Suzumura ([1976], [1983]), Deb [1983] and Lahiri [1991a], all use Szpilrajn s theorem, to establish conditions for various shades of rationalizability of choice functions. In recent times Donaldson and Weymark [1981], Duggan [1991] and Lahiri [1999b] use Szpilrajn s theorem to establish results similar to those available in Dushnik and Miller [1941]. In Lahiri [1999b] an independent proof of the theorem due to Dushnik and Miller is given which uses Zorn s Lemma explicitly. Szpilran s theorem is a deep theorem in its own right and the fact that it uses Zorn s Lemma, often makes it inaccessible to someone who has had no formal training in advanced set theory. This is because, Zorn s Lemma is proved using the axiom of choice and an intermediate theorem is the well-ordering theorem. Much of this is beyond the scope of an individual who has not studied advanced set theory. It would be considerably easier to grasp those aspects of rational choice theory where Zorn;s Lemma is applied, if there was simpler way to obtain the celebrated lemma. This is what we don in this note by replacing the axiom of choice by what we call chain axiom . The proof of Zorn s lemma which now does not require the axiom of choice or the well-ordering theorem, can be established quite easily using elementary set theory. Rational choice theory comprises a body of results which are sufficiently challenging in their own right. Szpilrajn s theorem is a major building block of rational choice theory. It does the subject or its students no good by making it unnecessarily inaccessible. By making the journey to a crucial result more arduous than it need be, we shift the focus of rational choice theory from the analysis of decision making to an important result in set theory. Our chain axiom may require a giant leap of faith for a set theorist. For us, however, it is a major step towards simplification.