On Strongly-equitable Social Welfare Orders Without the Axiom of Choice

TL;DR

This paper investigates the existence of strongly equitable social welfare orders without relying on the Axiom of Choice, showing they imply non-measurable sets but not ultrafilters.

Contribution

It proves that SEA orders imply non-Lebesgue-measurable sets but do not imply the existence of nonprincipal ultrafilters, clarifying their set-theoretic implications.

Findings

SEA orders imply non-measurable sets of reals.

SEA orders do not imply the existence of nonprincipal ultrafilters.

Answers open questions about the set-theoretic strength of SEA orders.

Abstract

Social welfare orders seek to combine the disparate preferences of an infinite sequence of generations into a single, societal preference order in some reasonably-equitable way. In [2] Dubey and Laguzzi study a type of social welfare order which they call SEA, for strongly equitable and (finitely) anonymous. They prove that the existence of a SEA order implies the existence of a set of reals which does not have the Baire property, and observe that a nonprincipal ultrafilter over $\mathbb{N}$ can be used to construct a SEA order. Questions arising in their work include whether the existence of a SEA order implies the existence of either a set of real numbers which is not Lebesgue-measurable or of a nonprincipal ultrafilter over $\mathbb{N}$. We answer both these questions, the solution to the second using the techniques of geometric set theory as set out by Larson and Zapletal in [11]. The outcome is that the existence of a SEA order does imply the existence of a set of reals which is not Lebesgue-measurable, and does not imply the existence of a nonprincipal ultrafilter on $\mathbb{N}$.