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:


1. $\Bbb P$ does not add sets of rank $\leq\alpha$.
2. $\Bbb P$ adds sets to $\kappa$.
3. $\Bbb P$ 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.



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> 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?
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Can we always add sets without collapsing cardinals or adding [very] bounded sets?
As was confirmed in the comments, if $0^\sharp$ does not exist then SCH must hold. If we now let $\lambda$ be the least $\beth$-fixed point above $\alpha$, we can consider the forcing $\mathbb{P}=\mathrm{Add}(\lambda^+,1)$. This has $\lambda^{++}$-cc precisely because SCH holds, ensuring $2^\lambda=\lambda^+$. It follows that $\mathbb{P}$ does not collapse cardinals and it clearly does not add sets at rank $\alpha$.


The situation is a bit weird, since all we really needed for the argument is that GCH (or SCH, I guess) holds unboundedly often. So what you are asking for will be true in models which are close to canonical inner models (i.e. things like $0^\sharp$, $0^\dagger$ etc. do not exist) but also in models with many large cardinals (at least a strongly compact).


