  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 $\mathsf{RCA}_0$.

Logic