Comparing and Contrasting Arrow's Impossibility Theorem and G\"odel's Incompleteness Theorem

Ori Livson; Mikhail Prokopenko

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arXiv · 2504.06589

Comparing and Contrasting Arrow's Impossibility Theorem and G\"odel's Incompleteness Theorem

Ori Livson·Mikhail Prokopenko

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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 various impossibility results that place limits on the potential fairness, rationality and consistency of social decision-making processes. However, a relationship between the fields' most seminal results: G\"odel's First Incompleteness Theorem of Formal Logic, and Arrow's Impossibility Theorem in Social Choice Theory is lacking. In this paper, we address this gap by introducing a general mathematical object called a Self-Reference System. Correspondences between the two theorems are formalised by abstracting well-known diagonalisation and fixed-point arguments, and consistency and completeness properties of provability predicates in the language of Self-Reference Systems. Nevertheless, we show that the mechanisms generating Arrovian impossibility and G\"odelian incompleteness have subtle differences.

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Ori Livson, Mikhail Prokopenko. 2025-04-09. Comparing and Contrasting Arrow's Impossibility Theorem and G\"odel's Incompleteness Theorem. https://arxiv.org/abs/2504.06589

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