Bounded Forcing Axioms and Baumgartner’s Conjecture

We study the spectrum of forcing notions between the iterations of σ-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of α-proper forcings for indecomposable countable ordinals α as well as the Axiom A forcings. We focus on the bounded forcing axioms for the hierarchy of αproper forcings. Following ideas of Shelah we separate them for distinct countable indecomposable ordinals. In the study of forcings completely embeddable into an iteration of σ-closed followed by ccc forcing, we present an equivalent characterization of this class in terms of Baumgartner’s Axiom A. This resolves a conjecture of Baumgartner from the 1980’s.