English

Arrow's theorem, ultrafilters, and reverse mathematics

Logic 2024-04-25 v2

Abstract

This paper initiates the reverse mathematics of social choice theory, studying Arrow's impossibility theorem and related results including Fishburn's possibility theorem and the Kirman--Sondermann theorem within the framework of reverse mathematics. We formalise fundamental notions of social choice theory in second-order arithmetic, yielding a definition of countable society which is tractable in RCA0\mathsf{RCA}_0. We then show that the Kirman--Sondermann analysis of social welfare functions can be carried out in RCA0\mathsf{RCA}_0. This approach yields a proof of Arrow's theorem in RCA0\mathsf{RCA}_0, and thus in PRA\mathrm{PRA}, since Arrow's theorem can be formalised as a Π10\Pi^0_1 sentence. Finally we show that Fishburn's possibility theorem for countable societies is equivalent to ACA0\mathsf{ACA}_0 over RCA0\mathsf{RCA}_0.

Keywords

Cite

@article{arxiv.2306.06471,
  title  = {Arrow's theorem, ultrafilters, and reverse mathematics},
  author = {Benedict Eastaugh},
  journal= {arXiv preprint arXiv:2306.06471},
  year   = {2024}
}

Comments

23 pages

R2 v1 2026-06-28T11:01:59.056Z