Forcing axioms and cardinal arithmetic

Abstract

We survey some recent results on the impact of strong forcing axioms such as the Proper Forcing Axiom PFA and Martin’s Maximum MM on cardinal arithmetic. We concentrate on three combinatorial principles which follow from strong forcing axioms: stationary set reflection, Moore’s Mapping Reflection Principle MRP and the P-ideal dichotomy introduced by Abraham and Todorcevic which play the key role in these results. We also discuss the structure of inner models of PFA and MM and present some open problems.