Recasting Arrow's Impossibility Theorem as Gödelian Incomputability

Abstract: Incomputability results in formal logic and the Theory of Computation (i.e., incompleteness and undecidability) have deep implications for the foundations of mathematics and computer science. Likewise, Social Choice Theory, a branch of Welfare Economics, contains several impossibility results that place limits on the potential fairness, rationality and consistency of social decision-making processes. A formal relationship between G\"odel's Incompleteness Theorems in formal logic, and Arrow's Impossibility Theorem in Social Choice Theory has long been conjectured. In this paper, we address this gap by bringing these two theories closer by introducing a general mathematical object called a Self-Reference System. Impossibility in Social Choice Theory is demonstrated to correspond to the impossibility of a Self-Reference System to interpret its own internal consistency. We also provide a proof of G\"odel's First Incompleteness Theorem in the same terms. Together, this recasts Arrow's Impossibility Theorem as incomputability in the G\"odelian sense. The incomputability results in both fields are shown to arise out of self-referential paradoxes. This is exemplified by a new proof of Arrow's Impossibility Theorem centred around Condorcet Paradoxes.