Martin's axioms, measurability and equiconsistency results

Abstract

Abstract We deal with the consistency strength of ZFC + variants of MA + suitable sets of reals are measurable (and/or Baire, and/or Ramsey). We improve the theorem of Harrington and Shelah [2] repairing the asymmetry between measure and category, obtaining also the same result for Ramsey. We then prove parallel theorems with weaker versions of Martin's axiom (MA( σ -centered), (MA( σ -linked)), , MA( K )), getting Mahlo, inaccessible and weakly compact cardinals respectively. We prove that if there exists r ∈ R such that and MA holds, then there exists a -selective filter on ω , and from the consistency of ZFC we build a model for ZFC + MA( I ) + every -set of reals is Lebesgue measurable, has the property of Baire and is Ramsey.