The Birth of Social Choice Theory from the Spirit of Mathematical Logic: Arrow’s Theorem in the Framework of Model Theory

  • Daniel Eckert
  • Frederik S. Herzberg


Arrow’s axiomatic foundation of social choice theory can be understood as an application of Tarski’s methodology of the deductive sciences—which is closely related to the latter’s foundational contribution to model theory. In this note we show in a model-theoretic framework how Arrow’s use of von Neumann and Morgenstern’s concept of winning coalitions allows to exploit the algebraic structures involved in preference aggregation; this approach entails an alternative indirect ultrafilter proof for Arrow’s dictatorship result. This link also connects Arrow’s seminal result to key developments and concepts in the history of model theory, notably ultraproducts and preservation results.


Arrow’s theorem Model theory Winning coalition Ultrafilter Ultraproduct Boolean algebra Homomorphism 

Mathematics Subject Classification

03C98 91B14 01A60 03C20 03G05 


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Public EconomicsUniversity of GrazGrazAustria
  2. 2.Center for Mathematical EconomicsBielefeld UniversityBielefeldGermany
  3. 3.Munich Center for Mathematical PhilosophyLudwig-Maximilians-UniversitätMunichGermany

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