Forcing axioms, approachability, and stationary set reflection

TL;DR

This paper explores the relationships between forcing axioms, stationary set reflection, and approachability, providing new results, clarifications, and simplifications in the context of set theory and forcing techniques.

Contribution

It establishes that certain implications between stationary reflection and Strong Chang's Conjecture are unidirectional, and simplifies existing results on forcing axioms and approachability.

Findings

An implication of Fuchino-Usuba cannot be reversed.

Strengthening and simplification of Krueger's results.

Identification of sharpness in related results.

Abstract

We prove a variety of theorems about stationary set reflection and concepts related to internal approachability. We prove that an implication of Fuchino-Usuba relating stationary reflection to a version of Strong Chang's Conjecture cannot be reversed; strengthen and simplify some results of Krueger about forcing axioms and approachability; and prove that some other related results of Krueger are sharp. We also adapt some ideas of Woodin to simplify and unify many arguments in the literature involving preservation of forcing axioms.