Approachable Free Subsets and Fine Structure Derived Scales

TL;DR

This paper explores the relationships between free subsets, the PCF conjecture, and large cardinal assumptions, establishing new forcing results and the existence of special scales in inner models.

Contribution

It introduces the Approachable Bounded Subset Property and demonstrates its consistency from certain large cardinal assumptions, also proving the existence of continuous tree-like scales in inner models.

Findings

The Approachable Bounded Subset Property can be forced from unbounded Mitchell orders.

Continuous tree-like scales exist on all products in canonical inner models.

The large cardinal hypothesis used is shown to be optimal.

Abstract

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal $\lambda$ for which the set of Mitchell orders $\{ o(\mu) \mid \mu < \lambda\}$ is unbounded in $\lambda$. Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.