I have a doubt about the reals that Cohen forcing adds to ground model.
If $\{c_i\}_{i\in I}$ are Cohen reals then $M[G]=M[\{c_i\}_{i\in I}]$ ?
where $G$ is a generic filter for Cohen forcing.
My attemp. each $p\in G$ is a partial function with domain in $[I\times\omega]^{<\omega}$. Then, if $\pi_1(dom(p))=\{i_1,\dots,i_n\}$ we have $p=\bigcup_{1\leq j\leq n}(c_{i_j}\upharpoonright dom(p))\in M[\{c_i\}_{i\in I}]$. So, $G\in M[\{c_i\}_{i\in I}]$. I'm right?.