Forcing a countable structure to belong to the ground model

Abstract

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