Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Azul Fatalini, Luke Serafin

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 $ω$. Funicular preorders include coordinatewise domination on $\mathbb{R}^ω$, 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.

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 containing no nonprincipal ultrafilters over . Funicular preorders include coordinatewise domination on , 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.
Paper Structure (14 sections, 15 theorems, 40 equations, 2 figures)

This paper contains 14 sections, 15 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Order relations
  4. General and descriptive set theory
  5. Cohen forcing
  6. Geometric set theory
  7. Suslin forcing and virtual conditions
  8. Placid, balanced, and $(3,2)$-balanced forcing
  9. Funicular preorders
  10. Forcing prelinearizations over the Solovay model
  11. Eliminating the inaccessible
  12. Free ultrafilters and $(m,n)$-amalgamation.
  13. Funicular orders without the inaccessible
  14. Outlook

Key Result

Theorem 2.2

Let $\mathrel{ { \raisebox{\dimeval{\ht\tw@-\height}}{\m@th{ \cr $\sqsubset$\cr { } \reflectbox{$\sim$}\cr { } }} } }$ and $\lesssim$ two preorders on a set $X$. If ${\mathrel{ { \raisebox{\dimeval{\ht\tw@-\height}}{\m@th{ \cr $\sqsubset$\cr { } \reflectbox{$\sim$}\cr { } }} } }} \subseteq {\les

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (43)

  • Definition 2.1
  • Theorem 2.2: $\mathsf{ZFC}$; Szpilrajn szpilrajn-extension and Hansson hansson-choice-preference
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8: larson-zapletal-GST
  • Definition 2.9: larson-zapletal-GST
  • Definition 2.10: larson-zapletal-GST
  • ...and 33 more