{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:55I7FQFHKG25LW2OTWDGPBVRBH","short_pith_number":"pith:55I7FQFH","schema_version":"1.0","canonical_sha256":"ef51f2c0a751b5d5db4e9d866786b109f3096fd2fd366e6f43a64b4099f29797","source":{"kind":"arxiv","id":"1410.1224","version":3},"attestation_state":"computed","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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1410.1224","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2014-10-05T23:23:16Z","cross_cats_sorted":[],"title_canon_sha256":"be23460d6950f6ca926b54a3df792760e27b9ba81f47cc8e820c81bc1be25ba0","abstract_canon_sha256":"9f8e0f901f3d64100abbd0610816f6baf349e24de945a715bda40fcc278dad46"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:19:50.366931Z","signature_b64":"CipJSpohsr/0q7cskVxyUJ729/xhgrV5uDzVNBZEpbZJTSaii7FXDwBg/aCkO6CZoi4u0rOfBdzIvMylmwmXBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef51f2c0a751b5d5db4e9d866786b109f3096fd2fd366e6f43a64b4099f29797","last_reissued_at":"2026-05-18T02:19:50.366287Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:19:50.366287Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $P \\Vdash \\dot{\\tau}\\cong\\check{M}$. 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