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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"},"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":"math/9809196","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.LO","submitted_at":"1998-09-15T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"8162e01aaa56d7d5c86fd6bac5cc518b94649b68f007feb36cdc461598af8f7a","abstract_canon_sha256":"41e5ecf56f2ada950fd829e715c604a6654f7d7eb626b4c1a94071fce714537e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:33.721218Z","signature_b64":"tp5Zl6xSC6Yt68x29BER8XC7QcJiP40ZQFCoE1L4Brm0dJPpYa0kG5smsTR/b/kqWhlqlwR/YAvJzL9faU+rBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3514b2de501f27fde23fcd2c5babe52002e3fd5ea330ae3669655d5e9063b5e7","last_reissued_at":"2026-05-18T01:05:33.720570Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:33.720570Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Hanf numbers of the infinitary order property","license":"","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Rami Grossberg, Saharon Shelah","submitted_at":"1998-09-15T00:00:00Z","abstract_excerpt":"We study several cardinal, and ordinal--valued functions that are relatives of Hanf numbers. 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