Funicular preorders can be prelinearized without nonprincipal ultrafilters over $\mathbb N$

Abstract

It is a consequence of the axiom of choice that every preorder can be extended to a total preorder while respecting the strict preorder relation. We call such an extension a prelinearization of the preorder and study the extent to which the axiom of choice is needed to construct prelinearizations. We isolate the class of funicular preorders, and show that these have prelinearizations in models of $\mathsf{ZF+DC}$ containing no nonprincipal ultrafilters over $\omega$. Funicular preorders include coordinatewise domination on $\mathbb{R}^\omega$, Turing reducibility, and various preorders arising in social choice theory. The relevant models are constructed first using tools from the geometric set theory of Larson and Zapletal, which requires an inaccessible cardinal, and then the inaccessible cardinal is eliminated using methods of amalgamation for Cohen reals.