"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:
1) $GCH$ fails everywhere,
2) there are no inaccessible cardinals,
3) there are no $\kappa-$Souslin trees,
4) Any non-trivial $c.c.c.$ forcing adds a real,
5) Any non-trivial $\kappa^+-$closed forcing notion collapses some cardinals.
Consistency of (1) is proved by Foreman-Woodin, (2) clearly can be consistent and the consistency of (4) is shown in "Forcing with c.c.c forcing notions, Cohen reals and Random reals".
My interest is in the consistency of (5). Let's consider the case $\kappa=\omega.$
Question 1. Is it consistent that any non-trivial $\aleph_1-$closed forcing notion collapses some cardinals?
The above question seems very difficult, and it is not difficult to show that for its consistency we need some very large cardinals. But maybe the following is simpler:
Question 2. Is it consistent that any non-trivial $\aleph_1-$closed forcing notion of size continuum collapses some cardinals? Does its consistency imply the existence of large cardinals?