We start with a model of $\sf ZFC$, $V$, for simplicity we can imagine that $V$ satisfies $\sf GCH$ or even $V=L$.


Let $\Bbb P$ be $\operatorname{Add}(\omega,\omega\_1)$, and let $G$ be a $V$-generic filter for $\Bbb P$. For $E\subseteq\omega\_1$, we will write $G\restriction E$ as the restriction of $G$ to the reals with coordinates in $E$.


By the fact that the Cohen forcing is ccc, if $x\in\Bbb R^{V[G]}$, then there is some $E\in[\omega\_1]^{\omega}\cap V$ such that $x\in V[G\restriction E]$. However, I was told there will be such $x$ without a minimal $E$.



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> What is an example for a real $x$ such that if $x\in V[G\restriction E]$, then there is some $E'\subsetneq E$ for which $x\in V[G\restriction E']$?
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Cohen real without a minimal support
Define $x \in 2^{\omega} \cap V[G]$ by $x(n) = $ the first bit of the $n$th Cohen real. Then for every $E \subseteq \omega\_1$, $E \in V$, we have $x \in V[G \upharpoonright E]$ iff $\omega \setminus E$ is finite. So there is no such minimal $E$ for $x$.


