Canonical fragments of the strong reflection principle

TL;DR

This paper isolates and analyzes fragments of Todorcevic's strong reflection principle related to various forcing classes, revealing their implications and limitations in set theory.

Contribution

It introduces a method to extract and study specific fragments of SRP corresponding to forcing classes, connecting them to forcing axioms and exploring their effects.

Findings

Forcing axioms imply the corresponding SRP fragments.

The stationary set preserving fragment of SRP equals SRP itself.

Subcomplete fragment of SRP captures major consequences of subcomplete forcing axiom.

Abstract

For an arbitrary forcing class $\Gamma$, the $\Gamma$-fragment of Todorcevic's strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for $\Gamma$ implies the $\Gamma$-fragment of SRP, (2) the stationary set preserving fragment of SRP is the full principle SRP, and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. Along the way, some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity are explored, and some limitations to the extent to which fragments of SRP may capture the effects of their corresponding forcing axioms are established.