  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.
