{"paper":{"title":"Forcing a countable structure to belong to the ground model","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Itay Kaplan, Saharon Shelah","submitted_at":"2014-10-05T23:23:16Z","abstract_excerpt":"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 is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1410.1224","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}