August 14
Gunter Fuchs,
CUNY
Canonical fragments of the strong reflection principle
I have been working over the past few years on the project of trying to improve our understanding of the forcing axiom for subcomplete forcing. The most compelling feature of this axiom is its consistency with the continuum hypothesis. On the other hand, it captures many of the major consequences of Martin's Maximum. It is a compelling feature of Martin's Maximum that many of its consequences filter through Todorcevic's Strong Reflection Principle SRP. SRP has some consequences that the subcomplete forcing axiom does not have, like the failure of CH and the saturation of the nonstationary ideal. It has been unclear until recently whether there is a version of SRP that relates to the subcomplete forcing axiom as the full SRP relates to Martin's Maximum, but it turned out that there is: I will detail how to associate in a canonical way to an arbitrary forcing class its corresponding fragment of SRP in such a way that (1) the forcing axiom for the forcing class implies its 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. I will describe how this association works, describe some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity, and say a little more about the extent of (3).