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In this model every isomorphism between $P(\\omega)/I$ and any other quotient $P(\\omega)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $\\omega$.\n  We can also assure that the dominating number is equal to $\\aleph_1$ and that $2^{\\aleph_1}>2^{\\aleph_0}$. Therefore the Calkin algebra has outer automorphisms while all automorphisms of $P(\\omega)/Fin$ are trivial.\n  P"},"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":"1112.3571","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2011-12-15T17:18:46Z","cross_cats_sorted":[],"title_canon_sha256":"e97e668fdd2d2550a4210dccc72ae311b835aa30164c7a1b38056e540f2ec02e","abstract_canon_sha256":"cd15749f9d42f3b9da0b9faed9af0d1e412673ff18f166af289d764eaf9c0b5a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:49.535799Z","signature_b64":"bGSr0MX3izK5U1+qmcQcNIGeNYd5Pjqz/wW1Hfa9kajmRMXnnxG3l6U+rKQX8kfkhpAIXfvLqaUAfNKrzOzEBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"067b7f2946b70053d65f04799e1745b0f12d0af6fcff9023c30a2abe513fabf0","last_reissued_at":"2026-05-18T03:40:49.535113Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:49.535113Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Trivial automorphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Ilijas Farah, Saharon Shelah","submitted_at":"2011-12-15T17:18:46Z","abstract_excerpt":"We prove that the statement `For all Borel ideals I and J on $\\omega$, every isomorphism between Boolean algebras $P(\\omega)/I$ and $P(\\omega)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every isomorphism between $P(\\omega)/I$ and any other quotient $P(\\omega)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $\\omega$.\n  We can also assure that the dominating number is equal to $\\aleph_1$ and that $2^{\\aleph_1}>2^{\\aleph_0}$. Therefore the Calkin algebra has outer automorphisms while all automorphisms of $P(\\omega)/Fin$ are trivial.\n  P"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3571","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1112.3571","created_at":"2026-05-18T03:40:49.535239+00:00"},{"alias_kind":"arxiv_version","alias_value":"1112.3571v2","created_at":"2026-05-18T03:40:49.535239+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1112.3571","created_at":"2026-05-18T03:40:49.535239+00:00"},{"alias_kind":"pith_short_12","alias_value":"AZ5X6KKGW4AF","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_16","alias_value":"AZ5X6KKGW4AFHVS7","created_at":"2026-05-18T12:26:24.575870+00:00"},{"alias_kind":"pith_short_8","alias_value":"AZ5X6KKG","created_at":"2026-05-18T12:26:24.575870+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AZ5X6KKGW4AFHVS7AR4Z4F2FWD","json":"https://pith.science/pith/AZ5X6KKGW4AFHVS7AR4Z4F2FWD.json","graph_json":"https://pith.science/api/pith-number/AZ5X6KKGW4AFHVS7AR4Z4F2FWD/graph.json","events_json":"https://pith.science/api/pith-number/AZ5X6KKGW4AFHVS7AR4Z4F2FWD/events.json","paper":"https://pith.science/paper/AZ5X6KKG"},"agent_actions":{"view_html":"https://pith.science/pith/AZ5X6KKGW4AFHVS7AR4Z4F2FWD","download_json":"https://pith.science/pith/AZ5X6KKGW4AFHVS7AR4Z4F2FWD.json","view_paper":"https://pith.science/paper/AZ5X6KKG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1112.3571&json=true","fetch_graph":"https://pith.science/api/pith-number/AZ5X6KKGW4AFHVS7AR4Z4F2FWD/graph.json","fetch_events":"https://pith.science/api/pith-number/AZ5X6KKGW4AFHVS7AR4Z4F2FWD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AZ5X6KKGW4AFHVS7AR4Z4F2FWD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AZ5X6KKGW4AFHVS7AR4Z4F2FWD/action/storage_attestation","attest_author":"https://pith.science/pith/AZ5X6KKGW4AFHVS7AR4Z4F2FWD/action/author_attestation","sign_citation":"https://pith.science/pith/AZ5X6KKGW4AFHVS7AR4Z4F2FWD/action/citation_signature","submit_replication":"https://pith.science/pith/AZ5X6KKGW4AFHVS7AR4Z4F2FWD/action/replication_record"}},"created_at":"2026-05-18T03:40:49.535239+00:00","updated_at":"2026-05-18T03:40:49.535239+00:00"}