Local reflections of choice

TL;DR

The paper investigates how small violations of choice, captured by a seed $S$ via $\mathsf{SVC}(S)$ (and $\mathsf{SVC}^+(S)$), enable local reflections of many global choice principles in symmetric extensions. By developing a framework that translates global forms such as $\mathsf{DC}_\lambda$, $\mathsf{AC}_\lambda$, $\mathsf{BPI}$, $\mathsf{KWP}_\alpha$, $\mathsf{PP}$, and related comparability and union properties into seed-centered local criteria, the author derives a suite of equivalences and implications that facilitate preservation proofs. The work also employs forcing constructions, notably Feferman-style models and Cohen’s model, to demonstrate sharpness and limitations of these local reflections, including the fact that an infinite seed can be partitioned into $\omega$ many non-empty pieces. Collectively, these results provide a robust toolkit for analyzing which choice principles survive or fail under symmetry-based extensions and offer deeper insight into the structure of choice without full AC. The findings have potential practical impact on set-theoretic constructions where targeted preservation or refutation of selection principles is desired, and they sharpen our understanding of the interplay between local seed properties and global choice.

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 $\mathsf{DC}$, $\mathsf{AC}_λ$, $\mathsf{PP}$, and other important forms of choice. As a consequence, we show that if $S$ is infinite then $S$ can be partitioned into $ω$ many non-empty subsets.

Theorems & Definitions (42)

• Theorem
• Theorem
• Proposition 3.1
• proof
• Claim 3.1.1
• proof : Proof of Claim
• Proposition 3.2
• proof
• Corollary 3.3
• proof