The grounded Martin's axiom

Abstract

AbstractWe introduce a variant of Martin's axiom, called the grounded Martin's axiom, or , which asserts that the universe is a c.c.c. forcing extension in which Martin's axiom holds for posets in the ground model. This principle already implies several of the combinatorial consequences of . The new axiom is shown to be consistent with the failure of and a singular continuum. We prove that is preserved in a strong way when adding a Cohen real and that adding a random real to a model of preserves (even though it destroys itself). We also consider the analogous variant of the proper forcing axiom.