FORCING AXIOMS, APPROACHABILITY, AND STATIONARY SET REFLECTION

Forcing indestructibility of set-theoretic axioms

Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Levy-Collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to א1. Later we give applications, among them the c...

Internal Approachability and Reflection

We prove that the Weak Reflection Principle does not imply that every stationary set reflects to an internally approachable set. We show that several variants of internal approachability, namely internally unbounded, internally stationary, and internally club, are not provably equivalent. Let λ ≥ ω2 be a regular cardinal, and consider an elementary substructure N ≺ H(λ) with size א1. Then N is ...

Forcing axioms and inner models of set theory

Simple Forcing Notions and Forcing Axioms

Forcing Axioms and Cardinal Arithmetic

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 Todorčević which play the key rol...