{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:O4CDA42SCCH2EJ2G4HPIZ6URLC","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e816611b9a20a7eceee7df51c143ed8b588bcbd5dc50c68fb9b48231596950ae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-11-23T09:02:11Z","title_canon_sha256":"24dbdccf5110ce783a57bd057c5df3522177b95615ae1138944423cdc5dadb0e"},"schema_version":"1.0","source":{"id":"1211.5441","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.5441","created_at":"2026-05-18T03:39:52Z"},{"alias_kind":"arxiv_version","alias_value":"1211.5441v2","created_at":"2026-05-18T03:39:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.5441","created_at":"2026-05-18T03:39:52Z"},{"alias_kind":"pith_short_12","alias_value":"O4CDA42SCCH2","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"O4CDA42SCCH2EJ2G","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"O4CDA42S","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:e7072d305d9f976913d8f3a8ffae306bcdacf5a130a8be88183217b05f56d5f7","target":"graph","created_at":"2026-05-18T03:39:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We indicate a way of distinguishing between structures, for which, two structures are said to be separable.Being separable implies being non-isomorphic. We show that for any first order theory $T$ in a countable language, if it has an uncountable set of countable models that are pairwise separable, then actually it has such a set of size $2^{\\aleph_0}$. Our result follows trivially assuming the Continuum Hypothesis ($CH$). We work here in $ZFC$ (only without $CH$).","authors_text":"Mohammad Assem","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-11-23T09:02:11Z","title":"Separating Models by Formulas and the Number of Countable Models"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.5441","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3555e66796569384b4402ce1d6a70c8ef8f44ea0d8e454cad5868170e6ea677b","target":"record","created_at":"2026-05-18T03:39:52Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e816611b9a20a7eceee7df51c143ed8b588bcbd5dc50c68fb9b48231596950ae","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2012-11-23T09:02:11Z","title_canon_sha256":"24dbdccf5110ce783a57bd057c5df3522177b95615ae1138944423cdc5dadb0e"},"schema_version":"1.0","source":{"id":"1211.5441","kind":"arxiv","version":2}},"canonical_sha256":"7704307352108fa22746e1de8cfa9158996325c25b10314c4ca1354b395485b0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7704307352108fa22746e1de8cfa9158996325c25b10314c4ca1354b395485b0","first_computed_at":"2026-05-18T03:39:52.569272Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:39:52.569272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ebyJGwvSVrwigGw4agX5kQHQUdEs1oych4wt66el0lMS0LjqW6mqKSo3lmaKP2EkZ3lGc46hMZ9tWSjyg1DxAw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:39:52.569696Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.5441","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3555e66796569384b4402ce1d6a70c8ef8f44ea0d8e454cad5868170e6ea677b","sha256:e7072d305d9f976913d8f3a8ffae306bcdacf5a130a8be88183217b05f56d5f7"],"state_sha256":"59b3363332726a02982a9caf7f1b9444c384bb96ee918932b7bd4c3f8d6d8783"}