Recasting Arrow's Impossibility Theorem as G""odelian Incomputability

2025-04-09

AI Summary

This paper investigates the deep logical isomorphism between Arrow’s impossibility theorem and Gödel’s first incompleteness theorem. Method: Introducing the novel abstract framework of “self-referential systems,” it formalizes preference aggregation in social choice as a computability problem; leveraging formal logic modeling, recursive function theory, and axiomatic social choice theory, it provides a new proof of Arrow’s theorem grounded in Condorcet’s paradox and concurrently reconstructs Gödel’s first incompleteness theorem. Contribution/Results: The paper rigorously establishes that Arrow’s impossibility is logically equivalent to the undecidability of self-consistency within self-referential systems—i.e., a Gödelian form of uncomputability. Its core contribution is a cross-disciplinary unification of social choice theory and the foundations of mathematical logic, revealing that both domains fundamentally arise from undecidability induced by self-reference.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Preferences
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Computational social science

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.

Problem

• Linking Arrow's Impossibility Theorem to Gödelian incomputability
• Exploring self-reference systems in social choice and logic
• Demonstrating paradoxes as the root of incomputability results

Innovation

• Introducing Self-Reference System for formal connection
• Linking Arrow's Theorem to Gödelian incomputability
• Proving incomputability via self-referential paradoxes