Guessing models, trees, and cardinal arithmetic

TL;DR

This paper explores the connections between the Guessing Model Property, a consequence of the Proper Forcing Axiom, and various aspects of cardinal arithmetic, providing new implications and extending known results in set theory.

Contribution

It proves that a weakened Guessing Model Property implies Shelah's Strong Hypothesis and shows how it influences the size of the continuum, also extending results related to trees under forcing axioms.

Findings

A weakening of the Guessing Model Property implies Shelah's Strong Hypothesis.

The Guessing Model Property constrains the size of the continuum, making $2^{oldsymbol{ ext{ extomega}}_1}$ as small as possible relative to $2^ ext{ extomega}$.

Under certain forcing axioms, all trees of height and size $oldsymbol{ extomega}_1$ are B-special.

Abstract

Since being isolated by Viale and Weiss in 2009, the Guessing Model Property has emerged as a particularly prominent and powerful consequence of the Proper Forcing Axiom. In this paper, we investigate connections between variations of the Guessing Model Property and cardinal arithmetic, broadly construed. We improve upon results of Viale and Krueger by proving that a weakening of the Guessing Model Property implies Shelah's Strong Hypothesis. We also prove that, though the Guessing Model Property is known not to put an upper bound on the size of the continuum, it does imply that $2^{\omega_1}$ is as small as possible relative to the value of $2^\omega$. Building on work of Laver, we prove that, in the extension of any model of $\mathsf{PFA}$ by a measure algebra, every tree of height and size $\omega_1$ is B-special (a generalization of specialness introduced by Baumgartner that can also hold of trees with uncountable branches). Finally, we investigate the impact of forcing axioms for Suslin and almost Suslin trees on guessing model properties. In particular, we prove thatif $S$ is a Suslin tree, then the axioms $\mathsf{PFA}(S)$ and $\mathsf{PFA}(S)[S]$ imply the Guessing Model Property and the Indestructible Guessing Model Property, respectively, and, if $T^*$ is an almost Suslin Aronszajn tree, then the axiom $\mathsf{PFA}(T^*)$ implies the Indestructible Guessing Model Property. This answers a number of questions of Cox and Krueger.