Local reflections of choice

TL;DR

This paper investigates how small violations of choice principles reflect locally to properties of seed sets, providing characterizations useful for preservation proofs and demonstrating specific partition properties of infinite sets.

Contribution

It introduces the concept of local reflections of choice principles under small violations and applies it to various forms of choice, offering new insights into their local behavior.

Findings

Reflections of $ ext{DC}$, $ ext{AC}_ ext{lambda}$, $ ext{PP}$, and other choice principles.

If $S$ is infinite, it can be partitioned into $ ext{omega}$ many non-empty subsets.

Local properties of seed sets characterize the failure or preservation of choice principles.

Abstract

Under the assumption of small violations of choice with seed $S$ ($\mathsf{SVC}(S)$), the failure of many choice principles reflect to to local properties of $S$, which can be a helpful characterisation for preservation proofs. We demonstrate the reflections of $\mathsf{DC}$, $\mathsf{AC}_\lambda$, $\mathsf{PP}$, and other important forms of choice. As a consequence, we show that if $S$ is infinite then $S$ can be partitioned into $\omega$ many non-empty subsets.