ARROW'S THEOREM, ULTRAFILTERS, AND REVERSE MATHEMATICS

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 ${\mathsf {RCA}}_0$ .