{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:MXMDK2GJXSQX2PN75MJG22OW6F","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":"b712e2e5cbf1c90524d1ee7d0038107903c040d1be3c4f46241ed6c2ab3891ab","cross_cats_sorted":["cs.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-08-15T12:50:20Z","title_canon_sha256":"b60606f5306a1ebfd6559ed8aea67c880779a3ac82d71da4555812a95916411a"},"schema_version":"1.0","source":{"id":"1908.05677","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1908.05677","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"arxiv_version","alias_value":"1908.05677v4","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05677","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"pith_short_12","alias_value":"MXMDK2GJXSQX","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"pith_short_16","alias_value":"MXMDK2GJXSQX2PN7","created_at":"2026-07-05T01:11:41Z"},{"alias_kind":"pith_short_8","alias_value":"MXMDK2GJ","created_at":"2026-07-05T01:11:41Z"}],"graph_snapshots":[{"event_id":"sha256:f4413eea400f7c2de67cd0336391af03afadc38f2bc4c719b5030bbd95497c76","target":"graph","created_at":"2026-07-05T01:11:41Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1908.05677/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Turing's famous 'machine' framework provides an intuitively clear conception of 'computing with real numbers'. A recursive counterexample to a theorem shows that the theorem does not hold when restricted to computable objects. These counterexamples are often crucial in establishing reversals in the Reverse Mathematics program. All the previous is essentially limited to a language that can only express countable mathematics directly. The aim of this paper is to show that reversals and recursive counterexamples, countable in nature as they might be, directly yield new and interesting results abo","authors_text":"Sam Sanders","cross_cats":["cs.LO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-08-15T12:50:20Z","title":"Lifting countable to uncountable mathematics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05677","kind":"arxiv","version":4},"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":"sha256:8ebce216e0321591145e27a7cf49bc6aa7ab70554c0e64839d901a65d92764e6","target":"record","created_at":"2026-07-05T01:11:41Z","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":"b712e2e5cbf1c90524d1ee7d0038107903c040d1be3c4f46241ed6c2ab3891ab","cross_cats_sorted":["cs.LO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2019-08-15T12:50:20Z","title_canon_sha256":"b60606f5306a1ebfd6559ed8aea67c880779a3ac82d71da4555812a95916411a"},"schema_version":"1.0","source":{"id":"1908.05677","kind":"arxiv","version":4}},"canonical_sha256":"65d83568c9bca17d3dbfeb126d69d6f166d44d14061e482af04b8f169f19e28c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"65d83568c9bca17d3dbfeb126d69d6f166d44d14061e482af04b8f169f19e28c","first_computed_at":"2026-07-05T01:11:41.240431Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:11:41.240431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Z8uVHLwiTTLncBGUW8DofkTteyZaabhLW66BJFerympwFBiJ7iqj1bX4xgwOAA/fRoguv39jAiyC7tWUgipdBg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:11:41.240962Z","signed_message":"canonical_sha256_bytes"},"source_id":"1908.05677","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8ebce216e0321591145e27a7cf49bc6aa7ab70554c0e64839d901a65d92764e6","sha256:f4413eea400f7c2de67cd0336391af03afadc38f2bc4c719b5030bbd95497c76"],"state_sha256":"af353d5b06d3b62b18f192c50a665c30d6987efaa7efb40c36857a6aba17ae27"}