Bounded stationary reflection II

TL;DR

This paper explores models of set theory where bounded stationary reflection holds at various cardinals, demonstrating its consistency with other combinatorial properties and extending previous results in the area.

Contribution

It constructs models where bounded stationary reflection holds at successors of singular cardinals and at certain regular cardinals, even when approachability fails.

Findings

Bounded stationary reflection holds at the successor of every singular cardinal above _\u03c9.

Models are constructed where bounded stationary reflection holds at ^+ but approachability property fails.

The results extend the understanding of the consistency and interaction of stationary reflection with other set-theoretic properties.

Abstract

Bounded stationary reflection at a cardinal $\lambda$ is the assertion that every stationary subset of $\lambda$ reflects but there is a stationary subset of $\lambda$ that does not reflect at arbitrarily high cofinalities. We produce a variety of models in which bounded stationary reflection holds. These include models in which bounded stationary reflection holds at the successor of every singular cardinal $\mu > \aleph_\omega$ and models in which bounded stationary reflection holds at $\mu^+$ but the approachability property fails at $\mu$.