Topic Archive: hierarchies of forcing axioms
I will give an overview over several hierarchies of forcing axioms, with an emphasis on their versions for subcomplete forcing, but in the instances where the concepts are new, their versions for more established classes of forcing, such as proper forcing, are of interest as well. The hierarchies are the traditional one, reaching from the bounded to the unbounded forcing axiom (i.e., versions of Martin’s axiom for classes other than ccc forcing), a hierarchy of resurrection axioms (related to work of Tsaprounis), and (inspired by work of Bagaria, Schindler and Gitman) the “virtual” versions of these hierarchies: the weak bounded forcing axiom hierarchy and the virtual resurrection axiom hierarchy). I will talk about how the levels of these hierarchies are intertwined, in terms of implications or consistency strength. In many cases, I can provide exact consistency strength calculations, which build on techniques to “seal” square sequences, using subcomplete forcing, in the sense that no thread can be added without collapsing ω1. This idea goes back to Todorcevic, in the context of proper forcing (which is completely different from subcomplete forcing).