Stationary tower free homogeneously Suslin scales

TL;DR

This paper provides a new proof that the pointclass of ${< ext{lambda}}$-homogeneously Suslin sets has the scale property, avoiding stationary tower forcing, in the context of a limit of Woodin cardinals.

Contribution

It offers an alternative proof for the scale property of ${< ext{lambda}}$-homogeneously Suslin sets without using stationary tower forcing.

Findings

Confirmed the scale property for ${< ext{lambda}}$-homogeneously Suslin sets.

Provided a proof that bypasses stationary tower forcing techniques.

Enhanced understanding of descriptive set theory under large cardinal assumptions.

Abstract

Let $\lambda$ be a limit of Woodin cardinals. It was shown by the second author that the pointclass of ${<\lambda}$-homogeneously Suslin sets has the scale property. We give a new proof of this fact, which avoids the use of stationary tower forcing.