On Countable Stationary Towers

TL;DR

This paper explores properties of countable stationary towers, linking their precipitousness to regularity properties of sets of reals in $L( extbf R)$, without relying on strong large cardinal assumptions.

Contribution

It introduces the concept of semiprecipitousness and connects tower precipitousness with regularity properties in $L( extbf R)$, expanding understanding of stationary towers.

Findings

Precipitousness of towers implies regularity properties of sets of reals.

Introduces semiprecipitousness and explores its relation to precipitousness.

Shows that weakly compact height towers have significant regularity implications.

Abstract

In this paper, we investigate properties of countable stationary towers. We derive the regularity properties of sets of reals in $L(\mathbf R)$ from some properties of countable stationary towers without explicit use of strong large cardinals such as Woodin cardinals. We also introduce the notion of semiprecipitousness and investigate its relation to precipitousness and presaturation of countable stationary towers. We show that precipitousness of countable stationary towers of weakly compact height implies the regularity properties of sets of reals in $L(\mathbf R)$.