{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:U77M5J7BPUCHER4KCZN3CB5LSQ","short_pith_number":"pith:U77M5J7B","canonical_record":{"source":{"id":"1711.07952","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-11-21T18:24:41Z","cross_cats_sorted":[],"title_canon_sha256":"d3800f3794f30ce955df3349ca4eef0feaf7b6a9f37c17892e37d38c91569488","abstract_canon_sha256":"9b4492711b0f868e97f66de130e311c6a86304c18296840ba0960fa580e8bfe7"},"schema_version":"1.0"},"canonical_sha256":"a7fecea7e17d0472478a165bb107ab941c7fec7521212ad7d7ff09e6c93b6f37","source":{"kind":"arxiv","id":"1711.07952","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.07952","created_at":"2026-05-18T00:12:48Z"},{"alias_kind":"arxiv_version","alias_value":"1711.07952v2","created_at":"2026-05-18T00:12:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.07952","created_at":"2026-05-18T00:12:48Z"},{"alias_kind":"pith_short_12","alias_value":"U77M5J7BPUCH","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_16","alias_value":"U77M5J7BPUCHER4K","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_8","alias_value":"U77M5J7B","created_at":"2026-05-18T12:31:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:U77M5J7BPUCHER4KCZN3CB5LSQ","target":"record","payload":{"canonical_record":{"source":{"id":"1711.07952","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-11-21T18:24:41Z","cross_cats_sorted":[],"title_canon_sha256":"d3800f3794f30ce955df3349ca4eef0feaf7b6a9f37c17892e37d38c91569488","abstract_canon_sha256":"9b4492711b0f868e97f66de130e311c6a86304c18296840ba0960fa580e8bfe7"},"schema_version":"1.0"},"canonical_sha256":"a7fecea7e17d0472478a165bb107ab941c7fec7521212ad7d7ff09e6c93b6f37","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:48.586732Z","signature_b64":"YoHCaFAipsWyTCN9CuOHtSlRjlqbMlQOf0J+GnIPW5h+d+zDpnotJwEtan2dwBO7wNERhzARZex6BnCjYirYCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7fecea7e17d0472478a165bb107ab941c7fec7521212ad7d7ff09e6c93b6f37","last_reissued_at":"2026-05-18T00:12:48.586272Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:48.586272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1711.07952","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:12:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9hZyY+PjzeW8jcxB74cu0nklqVQuqjQatDdDUe1S+fQg1WIp2Lri02mjrsB2tielaX05/6fZBbq33fx4OdEGAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T12:58:43.850157Z"},"content_sha256":"4a3fbd08182a8e4c811785aa72918186b6a3ea940f72347ace881f0b582b5b83","schema_version":"1.0","event_id":"sha256:4a3fbd08182a8e4c811785aa72918186b6a3ea940f72347ace881f0b582b5b83"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:U77M5J7BPUCHER4KCZN3CB5LSQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The universal finite set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Joel David Hamkins, W. Hugh Woodin","submitted_at":"2017-11-21T18:24:41Z","abstract_excerpt":"We define a certain finite set in set theory $\\{x\\mid\\varphi(x)\\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition $\\varphi$ has complexity $\\Sigma_2$, so that any affirmative instance of it $\\varphi(x)$ is verified in any sufficiently large rank-initial segment of the universe $V_\\theta$; the set is empty in any transitive model and others; and if $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.07952","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:12:48Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xyRyUgVWNyiKhSfr5MMhe4tcxrxRzhv4S+UM+QtEN6qHRyNGLMw32AQdPB317vdLgRiaJ7IYUcZ/DV30N4/qAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T12:58:43.850521Z"},"content_sha256":"1e881bf2a202c4a25787bb9b43adbc2df53873f28fa375e2bde25af46012d446","schema_version":"1.0","event_id":"sha256:1e881bf2a202c4a25787bb9b43adbc2df53873f28fa375e2bde25af46012d446"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/U77M5J7BPUCHER4KCZN3CB5LSQ/bundle.json","state_url":"https://pith.science/pith/U77M5J7BPUCHER4KCZN3CB5LSQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/U77M5J7BPUCHER4KCZN3CB5LSQ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-02T12:58:43Z","links":{"resolver":"https://pith.science/pith/U77M5J7BPUCHER4KCZN3CB5LSQ","bundle":"https://pith.science/pith/U77M5J7BPUCHER4KCZN3CB5LSQ/bundle.json","state":"https://pith.science/pith/U77M5J7BPUCHER4KCZN3CB5LSQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/U77M5J7BPUCHER4KCZN3CB5LSQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:U77M5J7BPUCHER4KCZN3CB5LSQ","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":"9b4492711b0f868e97f66de130e311c6a86304c18296840ba0960fa580e8bfe7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-11-21T18:24:41Z","title_canon_sha256":"d3800f3794f30ce955df3349ca4eef0feaf7b6a9f37c17892e37d38c91569488"},"schema_version":"1.0","source":{"id":"1711.07952","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1711.07952","created_at":"2026-05-18T00:12:48Z"},{"alias_kind":"arxiv_version","alias_value":"1711.07952v2","created_at":"2026-05-18T00:12:48Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.07952","created_at":"2026-05-18T00:12:48Z"},{"alias_kind":"pith_short_12","alias_value":"U77M5J7BPUCH","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_16","alias_value":"U77M5J7BPUCHER4K","created_at":"2026-05-18T12:31:46Z"},{"alias_kind":"pith_short_8","alias_value":"U77M5J7B","created_at":"2026-05-18T12:31:46Z"}],"graph_snapshots":[{"event_id":"sha256:1e881bf2a202c4a25787bb9b43adbc2df53873f28fa375e2bde25af46012d446","target":"graph","created_at":"2026-05-18T00:12:48Z","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 define a certain finite set in set theory $\\{x\\mid\\varphi(x)\\}$ and prove that it exhibits a universal extension property: it can be any desired particular finite set in the right set-theoretic universe and it can become successively any desired larger finite set in top-extensions of that universe. Specifically, ZFC proves the set is finite; the definition $\\varphi$ has complexity $\\Sigma_2$, so that any affirmative instance of it $\\varphi(x)$ is verified in any sufficiently large rank-initial segment of the universe $V_\\theta$; the set is empty in any transitive model and others; and if $\\","authors_text":"Joel David Hamkins, W. Hugh Woodin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-11-21T18:24:41Z","title":"The universal finite set"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.07952","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":"sha256:4a3fbd08182a8e4c811785aa72918186b6a3ea940f72347ace881f0b582b5b83","target":"record","created_at":"2026-05-18T00:12:48Z","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":"9b4492711b0f868e97f66de130e311c6a86304c18296840ba0960fa580e8bfe7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2017-11-21T18:24:41Z","title_canon_sha256":"d3800f3794f30ce955df3349ca4eef0feaf7b6a9f37c17892e37d38c91569488"},"schema_version":"1.0","source":{"id":"1711.07952","kind":"arxiv","version":2}},"canonical_sha256":"a7fecea7e17d0472478a165bb107ab941c7fec7521212ad7d7ff09e6c93b6f37","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a7fecea7e17d0472478a165bb107ab941c7fec7521212ad7d7ff09e6c93b6f37","first_computed_at":"2026-05-18T00:12:48.586272Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:12:48.586272Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YoHCaFAipsWyTCN9CuOHtSlRjlqbMlQOf0J+GnIPW5h+d+zDpnotJwEtan2dwBO7wNERhzARZex6BnCjYirYCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:12:48.586732Z","signed_message":"canonical_sha256_bytes"},"source_id":"1711.07952","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4a3fbd08182a8e4c811785aa72918186b6a3ea940f72347ace881f0b582b5b83","sha256:1e881bf2a202c4a25787bb9b43adbc2df53873f28fa375e2bde25af46012d446"],"state_sha256":"16c0677033e7b390f810cc31a1063c84fd264399509dd42c55c4c9846ed6c664"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/oNPxd1G2IbUtcJgfLPMROacKXgDtOCboPC24rTaQVsN2Q7+zuhxIhvEppwMeYFpGyZNZaCJhMoVScNwXi9XDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T12:58:43.853095Z","bundle_sha256":"0159328661e775dd5ecde3769e6dabc425e28fdb0970245a29fcacc44503b79d"}}