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 . We then show that the Kirman--Sondermann analysis of social welfare functions can be carried out in . This approach yields a proof of Arrow's theorem in , and thus in , since Arrow's theorem can be formalised as a sentence. Finally we show that Fishburn's possibility theorem for countable societies is equivalent to over .
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