Baumgartner's conjecture and bounded forcing axioms

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, the Axiom A forcings and forcings completely embeddable into an iteration of a σ-closed followed by a ccc forcing. For the latter class, we present an equivalent characterization in terms of Baumgartner’s Axiom A. This resolves a conjecture of Baumgartner from the 1980s. We also study the bounded forcing axioms for the hierarchy of αproper forcings. Following ideas of Shelah we separate them for distinct countable indecomposable ordinals.