Bounded forcing axioms and the continuum

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Abstract

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 Jensen's 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 BMM3 implies 21=ℵ2, as well as a stationary reflection principle which has many of the consequences of Martin's Maximum for objects of size 2. Finally, we give an example of a so-called boldface bounded forcing axiom implying 20=ℵ2.

MSC

03E35
03E50
03E05
03E65

Keywords

Bounded forcing axioms
Gaps
Open coloring axiom
The continuum
Boldface bounded forcing axioms

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