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The Ordering Principle and Higher Dependent Choice

Peter Holy, Jonathan Schilhan

TL;DR

The paper advances the study of ZF models with the Ordering Principle $OP$ and higher dependent choice $DC_{<\kappa}$ in the absence of $AC$ by developing a HAD-tower forcing and a symmetric extension framework. It generalizes Pincus' constructions to arbitrary regular uncountable cardinals $\kappa$ and $\lambda$, yielding models of $ZF+OP+DC_{<\lambda}+\lnot DC_\lambda$ (and related variants) and clarifying when these principles can hold without full choice. Central technical innovations include the introduction of hereditarily almost disjoint towers, a carefully structured forcing, and a symmetry analysis that preserves linear orders while destroying $AC$. The results provide a broad set of consistency examples and demonstrate how the ordering principle and higher DC can coexist with a failure of the axiom of choice in carefully controlled universes, with potential flexibility to adapt the method to other cardinals and configurations.

Abstract

We provide, for any regular uncountable cardinal $κ$, a new argument for Pincus' result on the consistency of $\mathrm{ZF}$ with the higher dependent choice principle $\mathrm{DC}_{<κ}$ and the ordering principle in the presence of a failure of the axiom of choice. We also generalise his methods and obtain these consistency results in a larger class of models.

The Ordering Principle and Higher Dependent Choice

TL;DR

The paper advances the study of ZF models with the Ordering Principle and higher dependent choice in the absence of by developing a HAD-tower forcing and a symmetric extension framework. It generalizes Pincus' constructions to arbitrary regular uncountable cardinals and , yielding models of (and related variants) and clarifying when these principles can hold without full choice. Central technical innovations include the introduction of hereditarily almost disjoint towers, a carefully structured forcing, and a symmetry analysis that preserves linear orders while destroying . The results provide a broad set of consistency examples and demonstrate how the ordering principle and higher DC can coexist with a failure of the axiom of choice in carefully controlled universes, with potential flexibility to adapt the method to other cardinals and configurations.

Abstract

We provide, for any regular uncountable cardinal , a new argument for Pincus' result on the consistency of with the higher dependent choice principle and the ordering principle in the presence of a failure of the axiom of choice. We also generalise his methods and obtain these consistency results in a larger class of models.
Paper Structure (8 sections, 14 theorems, 21 equations)

This paper contains 8 sections, 14 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Hereditarily almost disjoint towers
  3. Our forcing notion
  4. Our symmetric system
  5. The failure of AC
  6. Minimal Supports
  7. The Ordering Principle
  8. Higher dependent choice

Key Result

Lemma 5

If $p$ is a HAD tower, and $\alpha,\beta<\lambda$ with $(\alpha,\beta)\not\in t(p)$, then there is a HAD tower $q\le p$ such that

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • ...and 36 more