Hindman's Theorem in the hierarchy of Choice Principles

TL;DR

This paper investigates a version of Hindman's finite unions theorem within ZF set theory, establishing its logical strength and relation to weak choice principles, thereby clarifying its position in the hierarchy of choice axioms.

Contribution

It precisely characterizes the strength of a Hindman's theorem variant in ZF, relating it to classical weak choice principles and situating it within the hierarchy of choice axioms.

Findings

Hindman's theorem variant is provable without the Axiom of Choice in ZF.

The theorem's strength is equivalent to certain weak choice principles.

It clarifies the logical position of Hindman's theorem in the hierarchy of choice principles.

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 $\mathsf{AC}$.