Fragments of SRP, the nonstationary ideal, and diagonal reflection

Abstract

We investigate the relationships between fragments of SRP and diagonal reflection principles. It turns out that the subcomplete fragment of SRP, augmented with CH, implies that the nonstationary ideal on ${\omega }_1$ is not saturated, but that Larson's diagonal reflection principle $OS{R}_{{\omega }_2}$ holds. In contrast, it was known that the full principle SRP implies the saturation of the nonstationary ideal on ${\omega }_1$, but it does not imply $OS{R}_{{\omega }_2}$. We find several instances of this phenomenon: pairing (variations of) the subcomplete fragment of SRP with stronger failures of the nonstationary ideal on ${\omega }_1$ to be saturated allows us to conclude more instances of diagonal reflection principles, while boosting (the corresponding variations of) the full principle SRP with strong forms of the saturation of the nonstationary ideal allows us to produce models where these principles still hold, yet weaker versions of diagonal reflection fail.