A Characterization of the Boolean Prime Ideal Theorem in Terms of Forcing Notions

TL;DR

This paper explores the Boolean Prime Ideal theorem, providing equivalent formulations using forcing notions and demonstrating its consequences through Zorn's Lemma and Martin's Axiom-style arguments.

Contribution

It introduces new characterizations of the Boolean Prime Ideal theorem via forcing notions and filter intersections, linking it to well-known set-theoretic principles.

Findings

Equivalent formulations of the Boolean Prime Ideal theorem in terms of forcing notions

Demonstration of consequences using Zorn's Lemma and Martin's Axiom

Enhanced understanding of weak forms of the Axiom of Choice

Abstract

For certain weak versions of the Axiom of Choice (most notably, the Boolean Prime Ideal theorem), we obtain equivalent formulations in terms of partial orders, and filter-like objects within them intersecting certain dense sets or antichains. This allows us to prove some consequences of the Boolean Prime Ideal theorem using arguments in the style of those that use Zorn's Lemma, or Martin's Axiom.