{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:YOIKTRKNV76T7LDLW7UK2GVANJ","short_pith_number":"pith:YOIKTRKN","canonical_record":{"source":{"id":"1906.03930","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-06-10T14:17:19Z","cross_cats_sorted":["cs.FL"],"title_canon_sha256":"aca277a1f037f705f88ac3ca166d74967a21d58f2080eff5f34abe7e2aa7958a","abstract_canon_sha256":"72a9cf12a68f73e4e3cac27482c9c60025c1268a0b3b6e2744cb89849843e6cc"},"schema_version":"1.0"},"canonical_sha256":"c390a9c54daffd3fac6bb7e8ad1aa06a7468d292a7b1c3d078fb7de434876779","source":{"kind":"arxiv","id":"1906.03930","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.03930","created_at":"2026-05-17T23:43:44Z"},{"alias_kind":"arxiv_version","alias_value":"1906.03930v1","created_at":"2026-05-17T23:43:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.03930","created_at":"2026-05-17T23:43:44Z"},{"alias_kind":"pith_short_12","alias_value":"YOIKTRKNV76T","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"YOIKTRKNV76T7LDL","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"YOIKTRKN","created_at":"2026-05-18T12:33:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:YOIKTRKNV76T7LDLW7UK2GVANJ","target":"record","payload":{"canonical_record":{"source":{"id":"1906.03930","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-06-10T14:17:19Z","cross_cats_sorted":["cs.FL"],"title_canon_sha256":"aca277a1f037f705f88ac3ca166d74967a21d58f2080eff5f34abe7e2aa7958a","abstract_canon_sha256":"72a9cf12a68f73e4e3cac27482c9c60025c1268a0b3b6e2744cb89849843e6cc"},"schema_version":"1.0"},"canonical_sha256":"c390a9c54daffd3fac6bb7e8ad1aa06a7468d292a7b1c3d078fb7de434876779","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:44.501723Z","signature_b64":"Vhaj7+2LPfQsBZ/+gEnwKsVwoakFXikSFa9ZYUt+5jbGm4WhTe5zj8nFsohDb2YGTkDqwZdk/gTlYTXqKszhAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c390a9c54daffd3fac6bb7e8ad1aa06a7468d292a7b1c3d078fb7de434876779","last_reissued_at":"2026-05-17T23:43:44.500957Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:44.500957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.03930","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:43:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q2xNfpGM9yFe2pmA4pTs91eLSwrlF4fGkWDp/jmHeFLsLU+QQpaKgDbwEB0yGcBic7YFUjTYS7lX6yxoUf/qBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T18:32:03.381515Z"},"content_sha256":"6c25fb89ad31341836f8042b39cb51193945907b9619439d3459bdacbe1a95bd","schema_version":"1.0","event_id":"sha256:6c25fb89ad31341836f8042b39cb51193945907b9619439d3459bdacbe1a95bd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:YOIKTRKNV76T7LDLW7UK2GVANJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Formalization of the Axiom of Choice and its Equivalent Theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.FL"],"primary_cat":"cs.LO","authors_text":"Tianyu Sun, Wensheng Yu","submitted_at":"2019-06-10T14:17:19Z","abstract_excerpt":"In this paper, we describe the formalization of the axiom of choice and several of its famous equivalent theorems in Morse-Kelley set theory. These theorems include Tukey's lemma, the Hausdorff maximal principle, the maximal principle, Zermelo's postulate, Zorn's lemma and the well-ordering theorem. We prove the above theorems by the axiom of choice in turn, and finally prove the axiom of choice by Zermelo's postulate and the well-ordering theorem, thus completing the cyclic proof of equivalence between them. The proofs are checked formally using the Coq proof assistant in which Morse-Kelley s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.03930","kind":"arxiv","version":1},"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":"2026-05-17T23:43:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b3R7+oi1d953SNR5O/7/4OR9uMxJg6Ej4KieAo3pCyfH9n7B+b11eqVnUfi7aVwURWwy999MSJUozV/KZnNVCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T18:32:03.382161Z"},"content_sha256":"4ad37694a4c4ee9bc9337702bc0e4ea6d15cad892f534d2cfb0914c95f0ad658","schema_version":"1.0","event_id":"sha256:4ad37694a4c4ee9bc9337702bc0e4ea6d15cad892f534d2cfb0914c95f0ad658"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YOIKTRKNV76T7LDLW7UK2GVANJ/bundle.json","state_url":"https://pith.science/pith/YOIKTRKNV76T7LDLW7UK2GVANJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YOIKTRKNV76T7LDLW7UK2GVANJ/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":"2026-05-30T18:32:03Z","links":{"resolver":"https://pith.science/pith/YOIKTRKNV76T7LDLW7UK2GVANJ","bundle":"https://pith.science/pith/YOIKTRKNV76T7LDLW7UK2GVANJ/bundle.json","state":"https://pith.science/pith/YOIKTRKNV76T7LDLW7UK2GVANJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YOIKTRKNV76T7LDLW7UK2GVANJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:YOIKTRKNV76T7LDLW7UK2GVANJ","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":"72a9cf12a68f73e4e3cac27482c9c60025c1268a0b3b6e2744cb89849843e6cc","cross_cats_sorted":["cs.FL"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-06-10T14:17:19Z","title_canon_sha256":"aca277a1f037f705f88ac3ca166d74967a21d58f2080eff5f34abe7e2aa7958a"},"schema_version":"1.0","source":{"id":"1906.03930","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.03930","created_at":"2026-05-17T23:43:44Z"},{"alias_kind":"arxiv_version","alias_value":"1906.03930v1","created_at":"2026-05-17T23:43:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.03930","created_at":"2026-05-17T23:43:44Z"},{"alias_kind":"pith_short_12","alias_value":"YOIKTRKNV76T","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"YOIKTRKNV76T7LDL","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"YOIKTRKN","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:4ad37694a4c4ee9bc9337702bc0e4ea6d15cad892f534d2cfb0914c95f0ad658","target":"graph","created_at":"2026-05-17T23:43:44Z","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":"In this paper, we describe the formalization of the axiom of choice and several of its famous equivalent theorems in Morse-Kelley set theory. These theorems include Tukey's lemma, the Hausdorff maximal principle, the maximal principle, Zermelo's postulate, Zorn's lemma and the well-ordering theorem. We prove the above theorems by the axiom of choice in turn, and finally prove the axiom of choice by Zermelo's postulate and the well-ordering theorem, thus completing the cyclic proof of equivalence between them. The proofs are checked formally using the Coq proof assistant in which Morse-Kelley s","authors_text":"Tianyu Sun, Wensheng Yu","cross_cats":["cs.FL"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-06-10T14:17:19Z","title":"Formalization of the Axiom of Choice and its Equivalent Theorems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.03930","kind":"arxiv","version":1},"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":"sha256:6c25fb89ad31341836f8042b39cb51193945907b9619439d3459bdacbe1a95bd","target":"record","created_at":"2026-05-17T23:43:44Z","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":"72a9cf12a68f73e4e3cac27482c9c60025c1268a0b3b6e2744cb89849843e6cc","cross_cats_sorted":["cs.FL"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.LO","submitted_at":"2019-06-10T14:17:19Z","title_canon_sha256":"aca277a1f037f705f88ac3ca166d74967a21d58f2080eff5f34abe7e2aa7958a"},"schema_version":"1.0","source":{"id":"1906.03930","kind":"arxiv","version":1}},"canonical_sha256":"c390a9c54daffd3fac6bb7e8ad1aa06a7468d292a7b1c3d078fb7de434876779","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c390a9c54daffd3fac6bb7e8ad1aa06a7468d292a7b1c3d078fb7de434876779","first_computed_at":"2026-05-17T23:43:44.500957Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:43:44.500957Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Vhaj7+2LPfQsBZ/+gEnwKsVwoakFXikSFa9ZYUt+5jbGm4WhTe5zj8nFsohDb2YGTkDqwZdk/gTlYTXqKszhAg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:43:44.501723Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.03930","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c25fb89ad31341836f8042b39cb51193945907b9619439d3459bdacbe1a95bd","sha256:4ad37694a4c4ee9bc9337702bc0e4ea6d15cad892f534d2cfb0914c95f0ad658"],"state_sha256":"93375a2368d854f77ed0cfe7475025f9a42f37a7e3659432e7a8981d7840a32c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3gvAugWQ1X/AxjTAx7k9z/YVk/qbIOscA6TWaAYzfZYUfdLWNUI68Pna2EOVVCmVdWBP7cemG0H2FtGklTa8Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T18:32:03.385519Z","bundle_sha256":"131605f3d22280251433d930d86682b61ea41142248306d13b3bab52e58a98c0"}}