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For example we look at (1) mu_{T}^*(gamma, kappa):= min {mu^* for all phi in L_{infinity, omega}, with rk(phi)< gamma, if T has the (phi, mu^*)-order property then there exists a formula phi'(x;y) in L_{kappa^+, omega}, such that for every chi >= kappa, T has the (phi', chi)-order property}; and (2) mu^*(gamma, kappa):= sup{mu_T^*(gamma, kappa)| T in L_{k","authors_text":"Rami Grossberg, Saharon Shelah","cross_cats":[],"headline":"","license":"","primary_cat":"math.LO","submitted_at":"1998-09-15T00:00:00Z","title":"On Hanf numbers of the infinitary order property"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9809196","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":"sha256:14a9210e825dc86759ae3ebfc7ae8d696e9e418aa6ee4d1cc059dd019eba855e","target":"record","created_at":"2026-05-18T01:05:33Z","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":"41e5ecf56f2ada950fd829e715c604a6654f7d7eb626b4c1a94071fce714537e","cross_cats_sorted":[],"license":"","primary_cat":"math.LO","submitted_at":"1998-09-15T00:00:00Z","title_canon_sha256":"8162e01aaa56d7d5c86fd6bac5cc518b94649b68f007feb36cdc461598af8f7a"},"schema_version":"1.0","source":{"id":"math/9809196","kind":"arxiv","version":1}},"canonical_sha256":"3514b2de501f27fde23fcd2c5babe52002e3fd5ea330ae3669655d5e9063b5e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3514b2de501f27fde23fcd2c5babe52002e3fd5ea330ae3669655d5e9063b5e7","first_computed_at":"2026-05-18T01:05:33.720570Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:33.720570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tp5Zl6xSC6Yt68x29BER8XC7QcJiP40ZQFCoE1L4Brm0dJPpYa0kG5smsTR/b/kqWhlqlwR/YAvJzL9faU+rBA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:33.721218Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9809196","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14a9210e825dc86759ae3ebfc7ae8d696e9e418aa6ee4d1cc059dd019eba855e","sha256:f3f500b71c6d1f8382110c3a39269cbf70dfd857d64a7d8418c1671ae024e40d"],"state_sha256":"b8cbf091d1a2fc3e146f723a0d39020a746af33e609107f4e35f98513d8256fe"}