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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