TITLE: Who proved "sets in every generic are already in the ground model?"
QUESTION [11 upvotes]: Suppose $\mathbb{P}$ is a notion of forcing in the ground model $V$, and $X$ is a set which is in $V[G]$ for every $\mathbb{P}$-generic filter $G$. Then $X\in V$ already, by a fairly simple (if tedious) argument.
I'm working on a paper in which this fact is mentioned, and I would like to cite it properly; however, I can't seem to find any citation for it, and I don't recall when I learned the argument. So my question is:

What is a citation for "if $X$ is in every forcing extension by some poset $\mathbb{P}$, then $X$ is already in the ground model?"

REPLY [5 votes]: Here is the computability-theoretic version: Suppose $G=G_0\oplus G_1$ is 1-generic and suppose $X$ is computable from both: so $X=\Phi_1^{G_0}=\Phi_2^{G_1}$. Then also $X=\Phi^{G_0}=\Phi^{G_1}$ where (Posner's trick) we define $\Phi$ by: noneffectively pick a number $n_0$ with $G_0(n_0)\ne G_1(n_0)$ and use this to decide whether to run $\Phi_1$ or $\Phi_2$.
The set $S := \{\sigma = \sigma_0\oplus\sigma_1: \Phi^{\sigma_0}(n)\downarrow \ne \Phi^{\sigma_1}(n)\,\exists n\}$ is $\Sigma^0_1$ and by assumption $G$ does not meet it, i.e., $\sigma\in S$ implies $\sigma$ is not a prefix of $G$. Then by 1-genericity there is a prefix of $G$, $\tau$, such that no extension of $\tau$ lands in $S$. Then $X$ is computable. Indeed, given $n$ we may search for any extension of $\tau$ making $\Phi^{\tau_0}(n)$ converge; then $X(n)=\Phi^{\tau_0}(n)$.