Sh:1054

• Kaplan, I., & Shelah, S. (2016). Forcing a countable structure to belong to the ground model. MLQ Math. Log. Q., 62(6), 530–546. arXiv: 1410.1224 DOI: 10.1002/malq.201400094 MR: 3601093
• Abstract:
  Suppose that P is a forcing notion, L is a language (in \mathbb{V}), \dot{\tau} a P-name such that P forces “\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}\left[G\right]= \dot{\tau}\left[G_{1}\right] and \dot{\tau}_{2}\left[G\right]=\dot{\tau}\left[G_{2}\right]. Zapletal asked whether or not P\times P forces \dot{\tau}_{1} \cong\dot{\tau}_{2} implies that there is some M\in\mathbb{V} such that P forces \dot{\tau}\cong\check{M}. We answer this negatively and discuss related issues.
• published version (17p)

@article{Sh:1054,
 author = {Kaplan, Itay and Shelah, Saharon},
 title = {{Forcing a countable structure to belong to the ground model}},
 journal = {MLQ Math. Log. Q.},
 fjournal = {MLQ. Mathematical Logic Quarterly},
 volume = {62},
 number = {6},
 year = {2016},
 pages = {530--546},
 issn = {0942-5616},
 mrnumber = {3601093},
 mrclass = {03C95 (03C45 03C55 03E40)},
 doi = {10.1002/malq.201400094},
 note = {\href{https://arxiv.org/abs/1410.1224}{arXiv: 1410.1224}},
 arxiv_number = {1410.1224}
}