  Suppose that $P$ is a forcing notion, $L$ is a language (in $V$),
$\dot{\tau}$ a $P$-name such that $P\Vdash$ "$\dot{\tau}$ is a countable
$L$-structure". In the product $P\times P$, there are names
$\dot{\tau_{1}},\dot{\tau_{2}}$ such that for any generic filter $G=G_{1}\times
G_{2}$ over $P\times P$, $\dot{\tau}_{1}[G]=\dot{\tau}[G_{1}]$ and
$\dot{\tau}_{2}[G]=\dot{\tau}[G_{2}]$. Zapletal asked whether or not $P \times
P \Vdash \dot{\tau}_{1}\cong\dot{\tau}_{2}$ implies that there is some $M\in V$
such that $P \Vdash \dot{\tau}\cong\check{M}$. We answer this negatively and
discuss related issues.

Logic