This article contains a fundamental error in the limit case of the proof of properness of the relevant forcing notion. More specifically, the main problems in this proof stem from the fact that, in the construction in this paper, there is a priori no way to ensure isomorphism, at the level of the predicates involved, of the models obtained via the relevant copying. We renounce our original claim of having solved the problem of forcing Measuring together with 2 0 > 2 .

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 Baumgartnerʼs Axiom A. This resolves a conjecture of Baumgartner from the 1980s. We also study the bounded forcing axioms for the hierarchy of α-proper forcings. Following ideas of Shelah we separate them for distinct countable indecomposable ordinals.