Specialising Trees With Small Approximations II

TL;DR

This paper explores how certain ideals on regular cardinals imply a compactness property for trees and demonstrates the consistency of their existence with specific guessing models, extending previous specialization results.

Contribution

It establishes a link between ideals on regular cardinals and tree specialisability, and extends a main theorem to broader forcing conditions.

Findings

Existence of ideals implies compactness for specialisability of trees.

Consistency of ideals with stationarily many guessing models.

Extension of specialization theorem to broader forcing classes.

Abstract

We show that the existence of a well-known type of ideals on a regular cardinal $\lambda$ implies a compactness property concerning the specialisability of a tree of height $\lambda$ with no cofinal branches. We also use Neeman's method of side conditions to show that the existence of such ideals is consistent with stationarily many appropriate guessing models. These objects suffice to extend the main theorem of \cite{mhpr_spe}: one can generically specialise any branchless tree of height $\kappa^{++}$ with a ${<}\kappa$-closed, $\kappa^{+}$-proper, and $\kappa^{++}$-preserving forcing, which has the $\kappa^+$-approximation property.