Forcing a countable structure to belong to the ground model

TL;DR

The paper investigates whether a countable structure forced to be in the ground model must be isomorphic to a structure in the ground model, providing a negative answer to a question by Zapletal.

Contribution

It demonstrates that the assumption that a countable structure forced to be in the ground model is isomorphic to a ground model structure does not hold, addressing a question in forcing theory.

Findings

Counterexample to Zapletal's question

Shows that isomorphism in forcing extensions does not imply ground model isomorphism

Discusses related issues in forcing and model theory

Abstract

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.