Forcing a countable structure to belong to the ground model

Abstract

Suppose that $P$ is a forcing notion, $L$ is a language (in $V$), $\dot{\tau }$ a $P$-name such that $P⊩$ "$\dot{\tau }$ 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 $P\times P⊩{\dot{\tau }}_1\cong {\dot{\tau }}_2$ implies that there is some $M\in V$ such that $P⊩\dot{\tau }\cong \check{M}$. We answer this negatively and discuss related issues.