Let $\alpha$ be an ordinal, and let $a\subseteq\alpha$. Moreover, let $\beta>\alpha$ be an ordinal such that, in $L[a]$, $\alpha$ and $\beta$ have the same cardinality and such that $\beta$ is "reasonably closed", say $L_{\beta}[a]\models\text{ZF}^{-}$. Denote by $P_{\alpha}$ the forcing of finite partial functions from $\alpha$ to $\{0,1\}$, ordered by reverse inclusion. Does it follow that there is a subset $x\subseteq\alpha$ that is $P_{\alpha}$-generic over $L_{\beta}[a]$ (i.e. generic with respect to all dense subsets in $L_{\beta}[a]$) and such that $L[x]=L[a]$? In other words, can such "locally generic" sets encode everything? Login To add answer/comment