Incompatible bounded category forcing axioms

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...

Bounded forcing axioms and the continuum

We show that bounded forcing axioms (for instance, the Bounded Proper Forcing Axiom and the Bounded Semiproper Forcing Axiom) are consistent with the existence of (!2; !2)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen’s combinatorial principles for L at the level !2, and therefore with the existence of an !2-Suslin tree. We also show that the axiom we ...

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 indec...

Woodin's axiom (*), bounded forcing axioms, and precipitous ideals on ω₁

If the Bounded Proper Forcing Axiom BPFA holds, then Mouse Reflection holds at א2 with respect to all mouse operators up to the level of Woodin cardinals in the next ZFC–model. This yields that if Woodin’s Pmax axiom (∗) holds, then BPFA implies that V is closed under the “Woodin-in-the-nextZFC–model” operator. We also discuss stronger Mouse Reflection principles which we show to follow from st...

Simple Forcing Notions and Forcing Axioms