Stationary reflection principles and two cardinal tree properties

TL;DR

This paper explores how stationary and semi-stationary set reflection principles influence set-theoretic properties, establishing their implications for the Singular Cardinal Hypothesis, square principles, and certain cardinal tree properties.

Contribution

It demonstrates that semi-stationary reflection implies key set-theoretic hypotheses and introduces new connections between reflection principles and cardinal tree properties.

Findings

Semi-stationary reflection implies the Singular Cardinal Hypothesis.

Semi-stationary reflection leads to the failure of weak square principles.

Stationary and semi-stationary reflection principles imply certain cardinal tree properties.

Abstract

We study consequences of stationary and semi-stationary set reflection. We show that the semi stationary reflection principle implies the Singular Cardinal Hypothesis, the failure of weak square principle, etc. We also consider two cardinal tree properties introduced recently by Weiss and prove that they follow from stationary and semi stationary set reflection augmented with a weak form of Martin's Axiom. We also show that there are some differences between the two reflection principles which suggest that stationary set reflection is analogous to supercompactness whereas semi-stationary set reflection is analogous to strong compactness.