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 (?,?)-gaps and thus do not imply the Open Coloring Axiom. They are also consistent with Jensen's combinatorial principles for L at the level ?, and therefore with the existence of an ?-Suslin tree. We also show that the axiom we call BMM implies ?=?, as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size ?. Finally, we give an example of a so-called boldface bounded forcing axiom implying 2=?.
- Bounded forcing axioms
- Open coloring axiom
- The continuum
- Boldface bounded forcing axioms