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 NI. Later we give applications, among them the consistency of MM with N-omega not being Jonsson which answers a question raised in the set theory meeting at Oberwolfach in 2005.
JOURNAL OF SYMBOLIC LOGIC
forcing axioms,transfer principles
Construction of the real numbers,Discrete mathematics,Kuratowski closure axioms,Axiom,Forcing (mathematics),Mathematics