Squares and narrow systems

TL;DR

This paper explores the relationships between narrow system properties, square principles, and forcing axioms, demonstrating consistency results and introducing strengthened square principles to analyze cofinal branches in narrow systems.

Contribution

It establishes new consistency results linking narrow systems, square principles, and forcing axioms, and introduces strengthened square principles to study cofinal branches.

Findings

Consistency of leph_{\u221e+1} with narrow system property and square principles.

Proper Forcing Axiom implies cofinal branches in countable width systems.

Existence of narrow systems of width leph_1 with no cofinal branch.

Abstract

A narrow system is a combinatorial object introduced by Magidor and Shelah in connection with work on the tree property at successors of singular cardinals. In analogy to the tree property, a cardinal $\kappa$ satisfies the \emph{narrow system property} if every narrow system of height $\kappa$ has a cofinal branch. In this paper, we study connections between the narrow system property, square principles, and forcing axioms. We prove, assuming large cardinals, both that it is consistent that $\aleph_{\omega+1}$ satisfies the narrow system property and $\square_{\aleph_{\omega}, < \aleph_{\omega}}$ holds and that it is consistent that every regular cardinal satisfies the narrow system property. We introduce natural strengthenings of classical square principles and show how they can be used to produce narrow systems with no cofinal branch. Finally, we show that the Proper Forcing Axiom implies that every narrow system of countable width has a cofinal branch but is consistent with the existence of a narrow system of width $\omega_1$ with no cofinal branch.