A prelinearization of a preorder is an extension to a prelinear order which preserves strict inequalities of the preorder. This notion generalizes the concept of linearizing a partial order, and arises naturally in the economic theory of social choice via what are called social welfare orders. It is known that prelinearizations of Borel preorders do not generally exist in the Solovay model, and we study the amount of choice needed to construct them. Specifically we want to know whether the existence of prelinearizations of particular Borel preorders implies the existence of nonprincipal ultrafilters on omega. This question was asked in the special case of two types of social welfare orders by Dubey and Laguzzi. We construct, for a given Borel preorder with the property that an element of the ground model may be interpolated between mutually generic elements of the underlying space, a model of ZF+DC in which the preorder admits a prelinearization and which contains no nonprincipal ultrafilter on omega. This is accomplished using the geometric set theory notion of (3,2)-balanced forcing, and in particular solves both problems of Dubey and Laguzzi about social welfare orders.
Joint work with Azul Fatalini, Justin Moore, and Paul Larson.