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 $\aleph_1$. Later we give applications, among them the consistency of ${\rm MM}$ with $\aleph_\omega$ not being Jonsson which answers a question raised during Oberwolfach 2005.
