A note on the modal logic of symmetric extensions

TL;DR

This paper explores the modal logic framework for symmetric extensions, a generalization of forcing, to analyze the multiverse of models with varying choice axioms, introducing the concept of choice-switches.

Contribution

It introduces the concept of choice-switches within the modal logic of symmetric extensions and analyzes their independence properties in the multiverse of models.

Findings

Choice-switches are introduced as a new concept.

Any independent system of choice-switches is not independent from standard independent systems of buttons.

The work extends the understanding of the structure of the multiverse of models with and without the axiom of choice.

Abstract

Taking symmetric extensions can be considered as a generalisation of forcing, which produces a richer multiverse of models with and without the axiom of choice. We can study the structure of this multiverse using modal logic. In particular, we define the concept of of choice-switches, and show any independent system of choice-switches is not itself independent from any standard example of an independent system of buttons.