The Set Theoretic Ambit Of Arrow's Theorem

Abstract

Set theoretic formulation of Arrow's theorem, viewedin light of a taxonomy of transitive relations,serves to unmask the theorem's understatedgenerality. Under the impress of the independenceof irrelevant alternatives, the antipode of ceteris paribus reasoning, a purported compilerfunction either breaches some other rationalitypremise or produces the effet Condorcet. Types of cycles, each the seeming handiwork of avirtual voter disdaining transitivity, arerigorously defined. Arrow's theorem erects adilemma between cyclic indecision anddictatorship. Maneuvers responsive theretoare explicable in set theoretic terms. None ofthese gambits rival in simplicity the unassistedescape of strict linear orderings, which, by virtueof the Arrow–Sen reflexivity premise, are notcaptured by the theorem. Yet these are therelations among whose n-tuples the effetCondorcet is most frequent. A generalization andstronger theorem encompasses these and all otherlinear orderings and total tierings. Revisions tothe Arrow–Sen definitions of `choice set' and`rationalization' similarly enable one to generalizeSen's demonstration that some rational choicefunction always exists. Similarly may onegeneralize Debreu's theorems establishing conditionsunder which a binary relation may be represented bya continuous real-valued order homomorphism.