Maximality Principles in Modal Logic and the Axiom of Choice

TL;DR

This paper explores the set-theoretic strength of maximality principles in modal and intuitionistic logic, establishing their equivalence to fundamental set-theoretic axioms like AC and BPI.

Contribution

It provides new formulations of maximality principles and proves their equivalence to key set-theoretic axioms, clarifying their logical strength.

Findings

Stronger maximality principles are equivalent to the Axiom of Choice.

Weaker principles are equivalent to the Boolean Prime Ideal Theorem.

Two formulations of each principle are presented and analyzed.

Abstract

We investigate the set-theoretic strength of several maximality principles that play an important role in the study of modal and intuitionistic logics. We focus on the well-known Fine and Esakia maximality principles, present two formulations of each, and show that the stronger formulations are equivalent to the Axiom of Choice (AC), while the weaker ones to the Boolean Prime Ideal Theorem (BPI).