On Strongly-equitable Social Welfare Orders Without the Axiom of Choice
Luke Serafin
TL;DR
The paper addresses whether the existence of strongly-equitable, finitely anonymous (SEA) social welfare orders forces the existence of a $nonprincipal$ ultrafilter on $\mathbb{N}$ or of a non-Lebesgue-measurable set. It develops a forcing-based, geometric-set-theory framework to separate these implications, showing that SEA orders necessarily imply the existence of a set of reals without the Lebesgue property, but do not entail a $nonprincipal$ ultrafilter (in models with no such ultrafilter). It provides explicit SEA-constructions from ultrafilters and from $\mathbb{E}_0/\mathbb{E}_1$-type linearizations, and develops a general prelinearization method for tranquil Borel preorders that can yield SEA-like orders without ultrafilters in ultrafilter-free extensions. Overall, the work clarifies the logical strength of SEA orders, connects social welfare concepts to descriptive-set-theory forcing, and opens avenues for extending prelinearization techniques to broader definable preorders.
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 $\mathbb{N}$.