Arrow's theorem, ultrafilters, and reverse mathematics

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 $\mathsf{RCA}_0$. We then show that the Kirman--Sondermann analysis of social welfare functions can be carried out in $\mathsf{RCA}_0$. This approach yields a proof of Arrow's theorem in $\mathsf{RCA}_0$, and thus in $\mathrm{PRA}$, since Arrow's theorem can be formalised as a $\Pi^0_1$ sentence. Finally we show that Fishburn's possibility theorem for countable societies is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$.