Bounded forcing axioms and the continuum Original Research Article

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 (ω2,ω2)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensenʹs combinatorial principles for L at the level ω2, and therefore with the existence of an ω2-Suslin tree. We also show that the axiom we call View the MathML source implies View the MathML source, as well as a stationary reflection principle which has many of the consequences of Martinʹs Maximum for objects of size View the MathML source. Finally, we give an example of a so-called boldface bounded forcing axiom implying View the MathML source.