Given a model of $\sf ZFC$, and an infinite ordinal $\alpha$. Can we prove that there is always a cardinal $\kappa$, and a forcing $\Bbb P$, such that:

- $\Bbb P$ does not add sets of rank $\leq\alpha$.
- $\Bbb P$ adds sets to $\kappa$.
- $\Bbb P$ does not collapse cardinals.

Any two of the three can be easily done. 1+2 can be done with $\operatorname{Add}(\kappa,1)$ which may collapse $\kappa^+$; 1+3 is satisfied by the trivial forcing; and 2+3 is satisfied by adding a single Cohen real.

Satisfying all three would violate Foreman's Maximality Principle (any nontrivial forcing adds a real or collapses cardinals). But the consistency of the principle is an open problem.

Assume that $V$ and $L$ are "close enough" (e.g. $0^\#$ does not exist, or $V$ is set generic over $L$, or something of this sort). Can we prove that such $\kappa$ and $\Bbb P$ exist?

1more comment