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Comparing and Contrasting Arrow's Impossibility Theorem and Gödel's Incompleteness Theorem

Ori Livson, Mikhail Prokopenko

TL;DR

The paper develops a general theory of Self-Reference Systems to formalize a deep correspondence between Gödel's Incompleteness Theorem and Arrow's Impossibility Theorem. By encoding arithmetic via Gödel numbering and social choice via extended preference codomains, the authors illuminate diagonalisation and fixed-point mechanisms that generate incompleteness and impossibility, respectively. They introduce embeddable Self-Reference Systems and an Abstract Diagonalisation Lemma, revealing how Gödel sentences correspond to contradictory preference cycles and how dictatorships alter (or eliminate) quasi-Gödelian behavior. While the overlap highlights fundamental structural parallels, the authors also identify essential differences in the underlying mechanisms, notably the role of fixed points and the implications of dictatorship. The framework promises broader cross-domain insights into computability, decision theory, and the design of consistent yet expressive formal systems across disciplines.

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: Gödel'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 Gödelian incompleteness have subtle differences.

Comparing and Contrasting Arrow's Impossibility Theorem and Gödel's Incompleteness Theorem

TL;DR

The paper develops a general theory of Self-Reference Systems to formalize a deep correspondence between Gödel's Incompleteness Theorem and Arrow's Impossibility Theorem. By encoding arithmetic via Gödel numbering and social choice via extended preference codomains, the authors illuminate diagonalisation and fixed-point mechanisms that generate incompleteness and impossibility, respectively. They introduce embeddable Self-Reference Systems and an Abstract Diagonalisation Lemma, revealing how Gödel sentences correspond to contradictory preference cycles and how dictatorships alter (or eliminate) quasi-Gödelian behavior. While the overlap highlights fundamental structural parallels, the authors also identify essential differences in the underlying mechanisms, notably the role of fixed points and the implications of dictatorship. The framework promises broader cross-domain insights into computability, decision theory, and the design of consistent yet expressive formal systems across disciplines.

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: Gödel'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 Gödelian incompleteness have subtle differences.

Paper Structure

This paper contains 21 sections, 29 theorems, 30 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Incomputability
  3. Arrow's Impossibility Theorem
  4. Summary of Results
  5. Background
  6. Gödel's Incompleteness Theorem
  7. Algebraic Logic
  8. Social Choice Theory
  9. Condorcet's Paradox
  10. Results
  11. Encodings in Arithmetic Logic and Social Choice Theory
  12. Self-Reference Systems
  13. The Fixed-Point Property
  14. Embeddable Self-Reference Systems
  15. The Abstract Diagonalisation Lemma
  16. ...and 6 more sections

Key Result

Lemma 2.1.1

For any predicate $Q(x)$ in a theory of Arithmetic Logic $\mathcal{T}$ there exists a sentence $\mathcal{C}$ such that $Q(\ulcorner C \urcorner)$ and $C$ are logically equivalent, i.e.: $\mathcal{T} \vdash Q(\ulcorner C \urcorner) \leftrightarrow C$.

Theorems & Definitions (88)

  • Lemma 2.1.1: The Diagonalisation Lemma
  • Theorem 2.1.2: Gödel's (First) Incompleteness Theorem
  • Definition 2.2.1: Lindenbaum Algebras
  • Definition 2.3.1
  • Theorem 2.3.2: Arrow's Impossibility Theorem
  • Definition 2.3.3
  • Definition 2.3.4
  • Definition 3.1.1
  • Proposition 3.1.1
  • proof
  • ...and 78 more