### Forcing a countable structure to belong to the ground model

by Kaplan and Shelah. [KpSh:1054]

Math Logic Quarterly, 2016

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

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