Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Lévy 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 consistency of MM with ℵω not being Jónsson which answers a question raised in the set theory meeting at Oberwolfach in 2005.
"Forcing indestructibility of set-theoretic axioms." J. Symbolic Logic 72 (1) 349 - 360, March 2007. https://doi.org/10.2178/jsl/1174668399