The following is a result which I know as a weak form of Jensen's coding lemma$^*$ (first published in the book "Coding the universe"; also see http://www.jstor.org/stable/2273986):
For any class of ordinals $A\subseteq ON$ with $V=L[A]$, there is a class forcing $\mathcal{P}$ such that $\Vdash_\mathcal{P}V=L[a], a\subseteq\omega$.
(The coding theorem says more than this, but I doubt that the claim above has a substantially easier proof.)
My question is:
- Can the class forcing $\mathcal{P}$ can assumed to be a set forcing, in the case when $A$ is a set?
I suspect this is true; however, the proof of the coding lemma is too difficult for me to untangle easily, so I'm not sure this is true.
$^*$ I've heard this called "Jensen's Coding Theorem," but the book is by Jensen and Beller & Welch; is this indeed due to Jensen, or all three of them?