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