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Hindman's Theorem in the hierarchy of Choice Principles

David J. Fernández-Bretón

Abstract

In the context of $\mathsf{ZF}$, we analyze a version of Hindman's finite unions theorem on infinite sets, which normally requires the Axiom of Choice to be proved. We establish the implication relations between this statement and various classical weak choice principles, thus precisely locating the strength of the statement as a weak form of the $\mathsf{AC}$.

Hindman's Theorem in the hierarchy of Choice Principles

Abstract

In the context of , we analyze a version of Hindman's finite unions theorem on infinite sets, which normally requires the Axiom of Choice to be proved. We establish the implication relations between this statement and various classical weak choice principles, thus precisely locating the strength of the statement as a weak form of the .
Paper Structure (10 sections, 25 theorems, 6 equations, 2 figures)

This paper contains 10 sections, 25 theorems, 6 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Some basic results
  3. Models of $\mathnormal{\mathsf{ZF}}$ and $\mathnormal{\mathsf{ZFA}}$
  4. Transferable statements and finiteness classes
  5. The truth-value of $\mathnormal{\mathsf{HT}}$ in the models
  6. $\mathnormal{\mathsf{HT}}$ and $\mathnormal{\mathsf{BPI}}$, $\mathnormal{\mathsf{OEP}}$, $\mathnormal{\mathsf{KW}}$, $\mathnormal{\mathsf{OP}}$, $\mathnormal{\mathsf{KL}}$, $\mathnormal{\mathsf{DC}}$, $\mathnormal{\mathsf{CC}}$, $\mathnormal{\mathrm{Fin}=\mathrm{D}\text{-}\mathrm{Fin}}$
  7. $\mathnormal{\mathsf{HT}}$ and various flavours of $\mathnormal{\mathsf{RT}}$
  8. $\mathnormal{\mathsf{HT}}$ and other choice principles
  9. $\mathnormal{\mathsf{HT}}$ vs. Form 82
  10. A weaker Boolean form of $\mathnormal{\mathsf{HT}}$

Key Result

Proposition 2.1

All of the statements $\mathnormal{\mathsf{HT}}(k)$, as $k\in\mathbb N\setminus\{1\}$ varies, are equivalent under $\mathnormal{\mathsf{ZF}}$.

Figures (2)

  • Figure 1: Implications between $\mathnormal{\mathsf{HT}}$ and other classical choice principles.
  • Figure 2: Enhanced diagram of implication relations, now including $\mathnormal{\mathsf{HT}}_2(k)$ among the other classical choice principles.

Theorems & Definitions (57)

  • Definition 1.1
  • Proposition 2.1
  • proof
  • Definition 2.1: brot-cao-fernandez, Definition 3.6 (3), cf. Definition 3.1
  • Proposition 2.2
  • proof
  • Corollary 2.1
  • proof
  • Proposition 2.3
  • proof
  • ...and 47 more