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Then ${\\mathcal M}$ has a minimal elementary end extension that codes exactly the same subsets of M that ${\\mathcal N}$ does iff every set that is $\\Pi_1^0$-definable in $({\\mathcal M},{\\mathfrak X})$ is the union of countably many sets that are $\\Sigma_1^0$-definable."},"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":"1512.06478","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2015-12-21T03:05:41Z","cross_cats_sorted":[],"title_canon_sha256":"f618d33eb65f1024b2f954e8642e7a40337eb37c51360c923d091b4a30eae755","abstract_canon_sha256":"0847e5e205b8b4fe7ccfbd050b07dcfabc368336e18c58ac8ab4851417cbcdee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:04:57.685069Z","signature_b64":"mPkLEivJZtlL4O/8R4HL1hdMHv3I8Z4GfdlaslJUxUaQzv1av6f5XL5kUYNXGCDNKxThWpt4nY8zuWh7ep54Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"206b452825af7c5c40f4f1b5d7979fb569812d1a8e03491a1b74226f1e8cb279","last_reissued_at":"2026-05-18T01:04:57.684508Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:04:57.684508Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Minimal elementary end extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"James H. 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